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Petersen graph
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In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring, although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by A. B. Kempe observed that its vertices can represent the ten lines of the Desargues configuration, donald Knuth states that the Petersen graph is a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general. The Petersen graph also makes an appearance in tropical geometry, the cone over the Petersen graph is naturally identified with the moduli space of five-pointed rational tropical curves. The Petersen graph is the complement of the graph of K5. As a Kneser graph of the form K G2 n −1, n −1 it is an example of an odd graph. Geometrically, the Petersen graph is the graph formed by the vertices and edges of the hemi-dodecahedron, that is, a dodecahedron with opposite points, lines and faces identified together. Any nonplanar graph has as minors either the complete graph K5, or the bipartite graph K3,3. The K5 minor can be formed by contracting the edges of a perfect matching, the K3,3 minor can be formed by deleting one vertex and contracting an edge incident to each neighbor of the deleted vertex. The most common and symmetric plane drawing of the Petersen graph, however, this is not the best drawing for minimizing crossings, there exists another drawing with only two crossings. Thus, the Petersen graph has crossing number 2, each edge in this drawing is crossed at most once, so the Petersen graph is 1-planar. On a torus the Petersen graph can be drawn without edge crossings, the Petersen graph can also be drawn in the plane in such a way that all the edges have equal length. That is, it is a unit distance graph, the simplest non-orientable surface on which the Petersen graph can be embedded without crossings is the projective plane. This is the embedding given by the construction of the Petersen graph. This construction forms a map and shows that the Petersen graph has non-orientable genus 1. The Petersen graph is strongly regular and it is also symmetric, meaning that it is edge transitive and vertex transitive. More strongly, it is 3-arc-transitive, every directed path in the Petersen graph can be transformed into every other such path by a symmetry of the graph
2.
Cubic graph
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In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a graph is a 3-regular graph. Cubic graphs are also called trivalent graphs, a bicubic graph is a cubic bipartite graph. In 1932, Ronald M. Foster began collecting examples of symmetric graphs. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number s such that two oriented paths of length s can be mapped to each other by exactly one symmetry of the graph. He showed that s is at most 5, and provided examples of graphs with each value of s from 1 to 5. Semi-symmetric cubic graphs include the Gray graph, the Ljubljana graph, the Frucht graph is one of the two smallest cubic graphs without any symmetries, it possesses only a single graph automorphism, the identity automorphism. According to Brooks theorem every connected cubic graph other than the complete graph K4 can be colored with at most three colors, according to Vizings theorem every cubic graph needs either three or four colors for an edge coloring. A 3-edge-coloring is known as a Tait coloring, and forms a partition of the edges of the graph into three perfect matchings, by Königs line coloring theorem every bicubic graph has a Tait coloring. The bridgeless cubic graphs that do not have a Tait coloring are known as snarks and they include the Petersen graph, Tietzes graph, the Blanuša snarks, the flower snark, the double-star snark, the Szekeres snark and the Watkins snark. There is a number of distinct snarks. Cubic graphs arise naturally in topology in several ways, for example, if one considers a graph to be a 1-dimensional CW complex, cubic graphs are generic in that most 1-cell attaching maps are disjoint from the 0-skeleton of the graph. Cubic graphs are formed as the graphs of simple polyhedra in three dimensions, polyhedra such as the regular dodecahedron with the property that three faces meet at every vertex. An arbitrary graph embedding on a surface may be represented as a cubic graph structure known as a graph-encoded map. In this structure, each vertex of a graph represents a flag of the embedding, a mutually incident triple of a vertex, edge. The three neighbors of each flag are the three flags that may be obtained from it by changing one of the members of this mutually incident triple, there has been much research on Hamiltonicity of cubic graphs. In 1880, P. G. Tait conjectured that every polyhedral graph has a Hamiltonian circuit. William Thomas Tutte provided a counter-example to Taits conjecture, the 46-vertex Tutte graph, in 1971, Tutte conjectured that all bicubic graphs are Hamiltonian
3.
Graph automorphism
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In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automorphism of a graph G = is a permutation σ of the vertex set V, such that the pair of vertices form an edge if and that is, it is a graph isomorphism from G to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs, the composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Fruchts theorem, all groups can be represented as the group of a connected graph – indeed. Constructing the automorphism group is at least as difficult as solving the graph isomorphism problem, for, G and H are isomorphic if and only if the disconnected graph formed by the disjoint union of graphs G and H has an automorphism that swaps the two components. In fact, just counting the automorphisms is polynomial-time equivalent to graph isomorphism The graph automorphism problem is the problem of testing whether a graph has a nontrivial automorphism and it belongs to the class NP of computational complexity. Similar to the isomorphism problem, it is unknown whether it has a polynomial time algorithm or it is NP-complete. There is a polynomial algorithm for solving the graph automorphism problem for graphs where vertex degrees are bounded by a constant. The graph automorphism problem is polynomial-time many-one reducible to the isomorphism problem. While no worst-case polynomial-time algorithms are known for the general Graph Automorphism problem, several open-source software tools are available for this task, including NAUTY, BLISS and SAUCY. SAUCY and BLISS are particularly efficient for sparse graphs, e. g. SAUCY processes some graphs with millions of vertices in mere seconds, however, BLISS and NAUTY can also produce Canonical Labeling, whereas SAUCY is currently optimized for solving Graph Automorphism. It also appears that the support of all generators is limited by a linear function of n. However, this has not been established for a fact, as of March 2012, molecular symmetry can predict or explain chemical properties. Several graph drawing researchers have investigated algorithms for drawing graphs in such a way that the automorphisms of the graph become visible as symmetries of the drawing. It is not always possible to display all symmetries of the graph simultaneously, so it may be necessary to choose which symmetries to display, several families of graphs are defined by having certain types of automorphisms, An asymmetric graph is an undirected graph without any nontrivial automorphisms. A vertex-transitive graph is a graph in which every vertex may be mapped by an automorphism into any other vertex. An edge-transitive graph is a graph in which every edge may be mapped by an automorphism into any other edge. A symmetric graph is a such that every pair of adjacent vertices may be mapped by an automorphism into any other pair of adjacent vertices
4.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
5.
Graph theory
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In mathematics graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, Graphs are one of the prime objects of study in discrete mathematics. Refer to the glossary of graph theory for basic definitions in graph theory, the following are some of the more basic ways of defining graphs and related mathematical structures. To avoid ambiguity, this type of graph may be described precisely as undirected, other senses of graph stem from different conceptions of the edge set. In one more generalized notion, V is a set together with a relation of incidence that associates with each two vertices. In another generalized notion, E is a multiset of unordered pairs of vertices, Many authors call this type of object a multigraph or pseudograph. All of these variants and others are described more fully below, the vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge, V and E are usually taken to be finite, and many of the well-known results are not true for infinite graphs because many of the arguments fail in the infinite case. The order of a graph is |V|, its number of vertices, the size of a graph is |E|, its number of edges. The degree or valency of a vertex is the number of edges that connect to it, for an edge, graph theorists usually use the somewhat shorter notation xy. Graphs can be used to model many types of relations and processes in physical, biological, social, Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the network is sometimes defined to mean a graph in which attributes are associated with the nodes and/or edges. In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the structure of a website can be represented by a directed graph, in which the vertices represent web pages. A similar approach can be taken to problems in media, travel, biology, computer chip design. The development of algorithms to handle graphs is therefore of major interest in computer science, the transformation of graphs is often formalized and represented by graph rewrite systems. Graph-theoretic methods, in forms, have proven particularly useful in linguistics. Traditionally, syntax and compositional semantics follow tree-based structures, whose power lies in the principle of compositionality
6.
Graph (discrete mathematics)
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In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense related. The objects correspond to mathematical abstractions called vertices and each of the pairs of vertices is called an edge. Typically, a graph is depicted in form as a set of dots for the vertices. Graphs are one of the objects of study in discrete mathematics, the edges may be directed or undirected. In contrast, if any edge from a person A to a person B corresponds to As admiring B, then this graph is directed, because admiration is not necessarily reciprocated. The former type of graph is called a graph and the edges are called undirected edges while the latter type of graph is called a directed graph. Graphs are the subject studied by graph theory. The word graph was first used in this sense by J. J. Sylvester in 1878, the following are some of the more basic ways of defining graphs and related mathematical structures. In one very common sense of the term, a graph is an ordered pair G = comprising a set V of vertices, nodes or points together with a set E of edges, arcs or lines, which are 2-element subsets of V. To avoid ambiguity, this type of graph may be described precisely as undirected, other senses of graph stem from different conceptions of the edge set. In one more general conception, E is a set together with a relation of incidence that associates with each two vertices. In another generalized notion, E is a multiset of unordered pairs of vertices, many authors call these types of object multigraphs or pseudographs. All of these variants and others are described more fully below, the vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge, V and E are usually taken to be finite, and many of the well-known results are not true for infinite graphs because many of the arguments fail in the infinite case. Moreover, V is often assumed to be non-empty, but E is allowed to be the empty set, the order of a graph is |V|, its number of vertices. The size of a graph is |E|, its number of edges, the degree or valency of a vertex is the number of edges that connect to it, where an edge that connects to the vertex at both ends is counted twice. For an edge, graph theorists usually use the shorter notation xy. As stated above, in different contexts it may be useful to refine the term graph with different degrees of generality, whenever it is necessary to draw a strict distinction, the following terms are used
7.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
8.
Group action
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In mathematics, an action of a group is a way of interpreting the elements of the group as acting on some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, actions of groups on vector spaces are called representations of the group. Some groups can be interpreted as acting on spaces in a canonical way, more generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions. A common way of specifying non-canonical actions is to describe a homomorphism φ from a group G to the group of symmetries of a set X. The action of an element g ∈ G on a point x ∈ X is assumed to be identical to the action of its image φ ∈ Sym on the point x. The homomorphism φ is also called the action of G. Thus, if G is a group and X is a set, if X has additional structure, then φ is only called an action if for each g ∈ G, the permutation φ preserves the structure of X. The abstraction provided by group actions is a one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them, in particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, the group G is said to act on X. The set X is called a G-set. In complete analogy, one can define a group action of G on X as an operation X × G → X mapping to x. g. =. h for all g, h in G and all x in X, for a left action h acts first and is followed by g, while for a right action g acts first and is followed by h. Because of the formula −1 = h−1g−1, one can construct an action from a right action by composing with the inverse operation of the group. Also, an action of a group G on X is the same thing as a left action of its opposite group Gop on X. It is thus sufficient to only consider left actions without any loss of generality. The trivial action of any group G on any set X is defined by g. x = x for all g in G and all x in X, that is, every group element induces the identity permutation on X. In every group G, left multiplication is an action of G on G, g. x = gx for all g, x in G
9.
Vertex-transitive graph
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In the mathematical field of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f, V → V such that f = v 2. In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices, a graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every graph is regular. However, not all graphs are symmetric, and not all regular graphs are vertex-transitive. Finite vertex-transitive graphs include the symmetric graphs, the finite Cayley graphs are also vertex-transitive, as are the vertices and edges of the Archimedean solids. Potočnik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices, although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs. The most famous example is the Petersen graph, but others can be constructed including the graphs of edge-transitive non-bipartite graphs with odd vertex degrees. The edge-connectivity of a graph is equal to the degree d. If the degree is 4 or less, or the graph is also edge-transitive, or the graph is a minimal Cayley graph, infinite vertex-transitive graphs include, infinite paths infinite regular trees, e. g. A well known conjecture stated that every infinite graph is quasi-isometric to a Cayley graph. A counterexample was proposed by Diestel and Leader in 2001, in 2005, Eskin, Fisher, and Whyte confirmed the counterexample. Edge-transitive graph Lovász conjecture Semi-symmetric graph Zero-symmetric graph Weisstein, Eric W. Vertex-transitive graph, a census of small connected cubic vertex-transitive graphs. Primož Potočnik, Pablo Spiga, Gabriel Verret,2012
10.
Edge-transitive graph
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In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2. In other words, a graph is edge-transitive if its automorphism group acts transitively upon its edges, Edge-transitive graphs include any complete bipartite graph K m, n, and any symmetric graph, such as the vertices and edges of the cube. Symmetric graphs are also vertex-transitive, but in general edge-transitive graphs need not be vertex-transitive, the Gray graph is an example of a graph which is edge-transitive but not vertex-transitive. All such graphs are bipartite, and hence can be colored with two colors. An edge-transitive graph that is regular, but not vertex-transitive, is called semi-symmetric. The Gray graph again provides an example, every edge-transitive graph that is not vertex-transitive must be bipartite and either semi-symmetric or biregular. Edge-transitive Weisstein, Eric W. Edge-transitive graph
11.
Semi-symmetric graph
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In the mathematical field of graph theory, a semi-symmetric graph is an undirected graph that is edge-transitive and regular, but not vertex-transitive. A semi-symmetric graph must be bipartite, and its automorphism group must act transitively on each of the two sets of the bipartition. For instance, in the diagram of the Folkman graph shown here, green vertices can not be mapped to red ones by any automorphism, but every two vertices of the same color are symmetric with each other. Semi-symmetric graphs were first studied E. Dauber, a student of F. Harary, in a paper, no longer available, titled On line- but not point-symmetric graphs. This was seen by Jon Folkman, whose paper, published in 1967, includes the smallest semi-symmetric graph, now known as the Folkman graph, the term semi-symmetric was first used by Klin et al. in a paper they published in 1978. The smallest cubic graph is the Gray graph on 54 vertices. It was first observed to be semi-symmetric by Bouwer and it was proven to be the smallest cubic semi-symmetric graph by Dragan Marušič and Aleksander Malnič. All the cubic semi-symmetric graphs on up to 768 vertices are known
12.
Regular graph
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In graph theory, a regular graph is a graph where each vertex has the same number of neighbors, i. e. every vertex has the same degree or valency. A regular directed graph must also satisfy the condition that the indegree and outdegree of each vertex are equal to each other. A regular graph with vertices of degree k is called a graph or regular graph of degree k. Also, from the Handshaking lemma, a graph of odd degree will contain even number of vertices. A 3-regular graph is known as a cubic graph, the smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices. The complete graph K m is strongly regular for any m, a theorem by Nash-Williams says that every k‑regular graph on 2k +1 vertices has a Hamiltonian cycle. It is well known that the necessary and sufficient conditions for a k regular graph of n to exist are that n ≥ k +1. In such case it is easy to construct regular graphs by considering appropriate parameters for circulant graphs, let A be the adjacency matrix of a graph. Then the graph is regular if and only if j = is an eigenvector of A and its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to j, so for such eigenvectors v =, a regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one. The only if direction is a consequence of the Perron–Frobenius theorem. There is also a criterion for regular and connected graphs, a graph is connected and regular if and only if the matrix of ones J, with J i j =1, is in the adjacency algebra of the graph. Let G be a graph with diameter D and eigenvalues of adjacency matrix k = λ0 > λ1 ≥ ⋯ ≥ λ n −1. If G is not bipartite, then D ≤ log log +1, Regular graphs may be generated by the GenReg program. Random regular graph Strongly regular graph Moore graph Cage graph Highly irregular graph Weisstein, Weisstein, Eric W. Strongly Regular Graph. GenReg software and data by Markus Meringer
13.
Distance-transitive graph
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A distance transitive graph is vertex transitive and symmetric as well as distance regular. A distance-transitive graph is interesting partly because it has an automorphism group. Some interesting finite groups are the groups of distance-transitive graphs. Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith and these are, Independently in 1969 a Russian group led by Georgy Adelson-Velsky showed that there exist graphs that are distance-regular but not distance-transitive. The only graph of this type with three is the 126-vertex Tutte 12-cage. The smallest distance-regular graph that is not distance-transitive is the Shrikhande graph, complete lists of distance-transitive graphs are known for some degrees larger than three, but the classification of distance-transitive graphs with arbitrarily large vertex degree remains open. The simplest asymptotic family of examples of graphs is the Hypercube graphs. Other families are the folded cube graphs and the rooks graphs. All three of these families have high degree. Early works Adelson-Velskii, G. M. Veĭsfeĭler, B, leman, A. A. Faradžev, I. A. An example of a graph which has no group of automorphisms, Doklady Akademii Nauk SSSR,185, 975–976. Biggs, Norman, Intersection matrices for linear graphs, Combinatorial Mathematics and its Applications, London, Academic Press, pp. 15–23, MR0285421. Biggs, Norman, Finite Groups of Automorphisms, London Mathematical Society Lecture Note Series,6, London & New York, Cambridge University Press, Biggs, N. L. Smith, D. H. On trivalent graphs, Bulletin of the London Mathematical Society,3, 155–158, doi,10. 1112/blms/3.2.155, Smith, D. H. Primitive and imprimitive graphs, The Quarterly Journal of Mathematics. Second Series,22, 551–557, doi,10. 1093/qmath/22.4.551, surveys Biggs, N. L. Distance-Transitive Graphs, Algebraic Graph Theory, Cambridge University Press, pp. 155–163, chapter 20. Van Bon, John, Finite primitive distance-transitive graphs, European Journal of Combinatorics,28, 517–532, doi,10. 1016/j. ejc.2005.04.014, MR2287450. Brouwer, A. E. Cohen, A. M. Neumaier, A. Distance-Transitive Graphs, Distance-Regular Graphs, New York, Springer-Verlag, pp. 214–234, chapter 7. Royle, G. Distance-Transitive Graphs, Algebraic Graph Theory, New York, Springer-Verlag, pp. 66–69, Ivanov, A. A. Distance-transitive graphs and their classification, in Faradžev, I. A. Ivanov, A. A. Klin, M. et al
14.
Strongly regular graph
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In graph theory, a strongly regular graph is defined as follows. Let G = be a graph with v vertices and degree k. G is said to be regular if there are also integers λ and μ such that. Every two non-adjacent vertices have μ common neighbours, a graph of this kind is sometimes said to be an srg. Strongly regular graphs were introduced by Raj Chandra Bose in 1963, the complement of an srg is also strongly regular. A strongly regular graph is a graph with diameter 2. Pick any node as the node, in Level 0. Then its k neighbor nodes lie in Level 1, and all other nodes lie in Level 2. Nodes in Level 1 are directly connected to the root, hence they must have λ other neighbors in common with the root, and these common neighbors must also be in Level 1. Since each node has degree k, there are k − λ −1 edges remaining for each Level 1 node to connect to nodes in Level 2, therefore, there are k × edges between Level 1 and Level 2. Nodes in Level 2 are not directly connected to the root, hence they must have μ common neighbors with the root, there are nodes in Level 2, and each is connected to μ nodes in Level 1. Therefore the number of edges between Level 1 and Level 2 is × μ, equating the two expressions for the edges between Level 1 and Level 2, the relation follows. Let I denote the identity matrix and let J denote the matrix whose entries all equal 1, the adjacency matrix A of a strongly regular graph satisfies two equations. First, A J = J A = k J, which is a restatement of the vertex degree requirement, incidentally. Second, A2 + A + I = μ J, the first term gives the number of 2-step paths from each vertex to all vertices, the second term the 1-step paths, and the third term the 0-step paths. For the vertex pairs connected by an edge, the equation reduces to the number of such 2-step paths being equal to λ. For the vertex pairs not directly connected by an edge, the equation reduces to the number of such 2-step paths being equal to μ, for the trivial self-pairs, the equation reduces to the degree being equal to k. Conversely, a graph which is not a complete or null graph whose adjacency matrix satisfies both of the conditions is a strongly regular graph
15.
Skew-symmetric graph
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Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs. One may use the property to extend σ to an orientation-reversing function on the edges of G. The transpose graph of G is the graph formed by reversing every edge of G, and σ defines a graph isomorphism from G to its transpose. A path or cycle in a graph is said to be regular if, for each vertex v of the path or cycle. Every directed path graph with an number of vertices is skew-symmetric. Similarly, a cycle graph is skew-symmetric if and only if it has an even number of vertices. In this case, the number of different mappings σ that realize the skew symmetry of the graph equals half the length of the cycle. Each vertex of the graph corresponds to two vertices of the skew-symmetric graph, and each edge of the polar graph corresponds to two edges of the skew-symmetric graph. A closely related concept is the graph of Edmonds & Johnson. For the correspondence between bidirected graphs and skew-symmetric graphs see Zaslavsky, Section 5, or Babenko, to form the double covering graph from a polar graph G, create for each vertex v of G two vertices v0 and v1, and let σ = v1 − i. For each edge e = of G, create two directed edges in the graph, one oriented from u to v and one oriented from v to u. The undirected edges at each vertex of the graph may be partitioned into two subsets according to which vertex of the polar graph they go out of and come in to. A regular path or cycle of a skew-symmetric graph corresponds to a path or cycle in the graph that uses at most one edge from each subset of edges at each of its vertices. By removing the edges of such a path from a matching. Similarly, cycles that alternate between matched and unmatched edges are of importance in weighted matching problems, as Goldberg & Karzanov showed, an alternating path or cycle in an undirected graph may be modeled as a regular path or cycle in a skew-symmetric directed graph. Goldberg & Karzanov generalized alternating path algorithms to show that the existence of a path between any two vertices of a skew-symmetric graph may be tested in linear time. If the length function is allowed to have negative lengths, the existence of a regular cycle may be tested in polynomial time. Along with the problems arising in matchings, skew-symmetric generalizations of the max-flow min-cut theorem have also been studied
16.
Biregular graph
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If the degree of the vertices in U is x and the degree of the vertices in V is y, then the graph is said to be -biregular. Every complete bipartite graph K a, b is -biregular, the rhombic dodecahedron is another example, it is -biregular. An -biregular graph G = must satisfy the equation x | U | = y | V |, every regular bipartite graph is also biregular. Every edge-transitive graph that is not also vertex-transitive must be biregular, in particular every edge-transitive graph is either regular or biregular. The Levi graphs of configurations are biregular, a biregular graph is the Levi graph of an configuration if
17.
Cayley graph
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In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayleys theorem and uses a specified, usually finite and it is a central tool in combinatorial and geometric group theory. Suppose that G is a group and S is a generating set, the Cayley graph Γ = Γ is a colored directed graph constructed as follows, Each element g of G is assigned a vertex, the vertex set V of Γ is identified with G. Each generator s of S is assigned a color c s, for any g ∈ G, s ∈ S, the vertices corresponding to the elements g and g s are joined by a directed edge of colour c s. Thus the edge set E consists of pairs of the form, in geometric group theory, the set S is usually assumed to be finite, symmetric and not containing the identity element of the group. In this case, the uncolored Cayley graph is a graph, its edges are not oriented. Suppose that G = Z is the cyclic group and the set S consists of the standard generator 1. Similarly, if G = Z n is the cyclic group of order n. More generally, the Cayley graphs of cyclic groups are exactly the circulant graphs. The Cayley graph of the product of groups is the cartesian product of the corresponding Cayley graphs. A Cayley graph of the dihedral group D4 on two generators a and b is depicted to the left, red arrows represent composition with a. Since b is self-inverse, the lines, which represent composition with b, are undirected. Therefore the graph is mixed, it has eight vertices, eight arrows, the Cayley table of the group D4 can be derived from the group presentation ⟨ a, b | a 4 = b 2 = e, a b = b a 3 ⟩. A different Cayley graph of Dih4 is shown on the right, B is still the horizontal reflection and represented by blue lines, c is a diagonal reflection and represented by green lines. As both reflections are self-inverse the Cayley graph on the right is completely undirected and this graph corresponds to the presentation ⟨ b, c | b 2 = c 2 = e, b c b c = c b c b ⟩. The Cayley graph of the group on two generators a, b corresponding to the set S = is depicted at the top of the article. Travelling along an edge to the right represents right multiplication by a, since the free group has no relations, the Cayley graph has no cycles. This Cayley graph is a key ingredient in the proof of the Banach–Tarski paradox, a Cayley graph of the discrete Heisenberg group is depicted to the right
18.
Zero-symmetric graph
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More precisely, it is a connected vertex-transitive cubic graph whose edges are partitioned into three different orbits by the automorphism group. In these graphs, for two vertices u and v, there is exactly one graph automorphism that takes u into v. The name for class of graphs was coined by R. M. Foster in a 1966 letter to H. S. M. Coxeter. The smallest zero-symmetric graph is a graph with 18 vertices. Among planar graphs, the truncated cuboctahedral and truncated icosidodecahedral graphs are also zero-symmetric and these examples are all bipartite graphs. However, there exist larger examples of graphs that are not bipartite. There are 97687 zero-symmetric graphs on up to 1280 vertices and these graphs form 89% of the cubic Cayley graphs and 88% of all connected vertex-transitive cubic graphs on the same number of vertices. All known finite connected zero-symmetric graphs contain a Hamiltonian cycle, and this is a special case of the Lovász conjecture that every finite connected vertex-transitive graph and every finite Cayley graph is Hamiltonian. Semi-symmetric graph, graphs that have symmetries between every two edges but not between two vertices
19.
Asymmetric graph
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In graph theory, a branch of mathematics, an undirected graph is called an asymmetric graph if it has no nontrivial symmetries. Formally, an automorphism of a graph is a permutation p of its vertices with the property that any two vertices u and v are adjacent if and only if p and p are adjacent. The identity mapping of a graph onto itself is always an automorphism, an asymmetric graph is a graph for which there are no other automorphisms. The smallest asymmetric non-trivial graphs have 6 vertices, the smallest asymmetric regular graphs have ten vertices, there exist ten-vertex asymmetric graphs that are 4-regular and 5-regular. One of the two smallest asymmetric cubic graphs is the twelve-vertex Frucht graph discovered in 1939, according to a strengthened version of Fruchts theorem, there are infinitely many asymmetric cubic graphs. The class of graphs is closed under complements, a graph G is asymmetric if. Any n-vertex asymmetric graph can be made symmetric by adding and removing a total of at most n/2 + o edges, the proportion of graphs on n vertices with nontrivial automorphism tends to zero as n grows, which is informally expressed as almost all finite graphs are asymmetric. In contrast, again informally, almost all graphs are symmetric. More specifically, countable infinite random graphs in the Erdős–Rényi model are, with probability 1, the smallest asymmetric tree has seven vertices, it consists of three paths of lengths 1,2, and 3, linked at a common endpoint. In contrast to the situation for graphs, almost all trees are symmetric
20.
Holt graph
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It is named after Peter G. Doyle and Derek F. Holt, who discovered the same graph independently in 1976 and 1981 respectively. The Holt Graph has diameter 3, radius 3 and girth 5, chromatic number 3 and it is also a 4-vertex-connected and a 4-edge-connected graph. It has an group of order 54 automorphisms. This is a group than a symmetric graph with the same number of vertices and edges would have. The graph drawing on the right highlights this, in that it lacks reflectional symmetry, the characteristic polynomial of the Holt graph is 644
21.
Bell Labs
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Nokia Bell Labs is an American research and scientific development company, owned by Finnish company Nokia. Its headquarters are located in Murray Hill, New Jersey, in addition to laboratories around the rest of the United States. The historic laboratory originated in the late 19th century as the Volta Laboratory, Bell Labs was also at one time a division of the American Telephone & Telegraph Company, half-owned through its Western Electric manufacturing subsidiary. Eight Nobel Prizes have been awarded for work completed at Bell Laboratories, in 1880, the French government awarded Alexander Graham Bell the Volta Prize of 50,000 francs, approximately US$10,000 at that time for the invention of the telephone. Bell used the award to fund the Volta Laboratory in Washington, D. C. in collaboration with Sumner Tainter, the laboratory is also variously known as the Volta Bureau, the Bell Carriage House, the Bell Laboratory and the Volta Laboratory. The laboratory focused on the analysis, recording, and transmission of sound, Bell used his considerable profits from the laboratory for further research and education to permit the diffusion of knowledge relating to the deaf. This resulted in the founding of the Volta Bureau c,1887, located at Bells fathers house at 1527 35th Street in Washington, D. C. where its carriage house became their headquarters in 1889. In 1893, Bell constructed a new building, close by at 1537 35th St. specifically to house the lab, the building was declared a National Historic Landmark in 1972. In 1884, the American Bell Telephone Company created the Mechanical Department from the Electrical, the first president of research was Frank B. Jewett, who stayed there until 1940, ownership of Bell Laboratories was evenly split between AT&T and the Western Electric Company. Its principal work was to plan, design, and support the equipment that Western Electric built for Bell System operating companies and this included everything from telephones, telephone exchange switches, and transmission equipment. Bell Labs also carried out consulting work for the Bell Telephone Company, a few workers were assigned to basic research, and this attracted much attention, especially since they produced several Nobel Prize winners. Until the 1940s, the principal locations were in and around the Bell Labs Building in New York City. Of these, Murray Hill and Crawford Hill remain in existence, the largest grouping of people in the company was in Illinois, at Naperville-Lisle, in the Chicago area, which had the largest concentration of employees prior to 2001. Since 2001, many of the locations have been scaled down or closed. The Holmdel site, a 1.9 million square foot structure set on 473 acres, was closed in 2007, the mirrored-glass building was designed by Eero Saarinen. In August 2013, Somerset Development bought the building, intending to redevelop it into a commercial and residential project. The prospects of success are clouded by the difficulty of readapting Saarinens design and by the current glut of aging, eight Nobel Prizes have been awarded for work completed at Bell Laboratories
22.
Girth (graph theory)
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In graph theory, the girth of a graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles, its girth is defined to be infinity, for example, a 4-cycle has girth 4. A grid has girth 4 as well, and a mesh has girth 3. A graph with four or more is triangle-free. A cubic graph of g that is as small as possible is known as a g-cage. The Petersen graph is the unique 5-cage, the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage, there may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices, the Balaban 10-cage, the Harries graph, paul Erdős was the first to prove the general result, using the probabilistic method. The odd girth and even girth of a graph are the lengths of a shortest odd cycle, the circumference of a graph is the length of the longest cycle, rather than the shortest. Thought of as the least length of a cycle, the girth admits natural generalisations as the 1-systole or higher systoles in systolic geometry. Girth is the concept to edge connectivity, in the sense that the girth of a planar graph is the edge connectivity of its dual graph. These concepts are unified in matroid theory by the girth of a matroid, for a graphic matroid, the matroid girth equals the girth of the underlying graph, while for a co-graphic matroid it equals the edge connectivity
23.
Complete graph
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In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a graph in which every pair of distinct vertices is connected by a pair of unique edges. Graph theory itself is dated as beginning with Leonhard Eulers 1736 work on the Seven Bridges of Königsberg. However, drawings of graphs, with their vertices placed on the points of a regular polygon, appeared already in the 13th century. Such a drawing is referred to as a mystic rose. The complete graph on n vertices is denoted by Kn, Kn has n/2 edges, and is a regular graph of degree n −1. All complete graphs are their own maximal cliques and they are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. The complement graph of a graph is an empty graph. If the edges of a graph are each given an orientation. The number of matchings of the graphs are given by the telephone numbers 1,1,2,4,10,26,76,232,764,2620,9496,35696,140152,568504,2390480,10349536,46206736. These numbers give the largest possible value of the Hosoya index for an n-vertex graph, the number of perfect matchings of the complete graph Kn is given by the double factorial. The crossing numbers up to K27 are known, with K28 requiring either 7233 or 7234 crossings, further values are collected by the Rectilinear Crossing Number project. Crossing numbers for Kn are 0,0,0,0,1,3,9,19,36,62,102,153,229,324,447,603,798,1029,1318,1657,2055,2528,3077,3699,4430,5250,6180. A complete graph with n nodes represents the edges of an -simplex, geometrically K3 forms the edge set of a triangle, K4 a tetrahedron, etc. The Császár polyhedron, a polyhedron with the topology of a torus, has the complete graph K7 as its skeleton. Every neighborly polytope in four or more dimensions also has a complete skeleton, k1 through K4 are all planar graphs. As part of the Petersen family, K6 plays a role as one of the forbidden minors for linkless embedding. In other words, and as Conway and Gordon proved, every embedding of K6 into three-dimensional space is intrinsically linked, Conway and Gordon also showed that any three-dimensional embedding of K7 contains a Hamiltonian cycle that is embedded in space as a nontrivial knot
24.
Complete bipartite graph
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Graph theory itself is typically dated as beginning with Leonhard Eulers 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, Llull himself had made similar drawings of complete graphs three centuries earlier. That is, it is a graph such that for every two vertices v1 ∈ V1 and v2 ∈ V2, v1v2 is an edge in E. A complete bipartite graph with partitions of size |V1|=m and |V2|=n, is denoted Km, n, for any k, K1, k is called a star. All complete bipartite graphs which are trees are stars, the graph K1,3 is called a claw, and is used to define the claw-free graphs. The graph K3,3 is called the utility graph and this usage comes from a standard mathematical puzzle in which three utilities must each be connected to three buildings, it is impossible to solve without crossings due to the nonplanarity of K3,3. Given a bipartite graph, testing whether it contains a complete bipartite subgraph Ki, a planar graph cannot contain K3,3 as a minor, an outerplanar graph cannot contain K3,2 as a minor. Conversely, every nonplanar graph contains either K3,3 or the complete graph K5 as a minor, Kn, n is a Moore graph and a -cage. The complete bipartite graphs Kn, n and Kn, n+1 have the possible number of edges among all triangle-free graphs with the same number of vertices. The complete bipartite graph Km, n has a vertex covering number of min, the complete bipartite graph Km, n has a maximum independent set of size max. The adjacency matrix of a bipartite graph Km, n has eigenvalues √, −√ and 0, with multiplicity 1,1. The Laplacian matrix of a bipartite graph Km, n has eigenvalues n+m, n, m. A complete bipartite graph Km, n has mn−1 nm−1 spanning trees, a complete bipartite graph Km, n has a maximum matching of size min. A complete bipartite graph Kn, n has a proper n-edge-coloring corresponding to a Latin square, every complete bipartite graph is a modular graph, every triple of vertices has a median that belongs to shortest paths between each pair of vertices
25.
Cube
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors
26.
Heawood graph
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In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood. The graph is cubic, and all cycles in the graph have six or more edges, every smaller cubic graph has shorter cycles, so this graph is the 6-cage, the smallest cubic graph of girth 6. It is a graph and therefore distance regular. There are 24 perfect matchings in the Heawood graph, for each matching, for instance, the figure shows the vertices of the graph placed on a cycle, with the internal diagonals of the cycle forming a matching. By subdividing the cycle edges into two matchings, we can partition the Heawood graph into three perfect matchings in eight different ways, every two perfect matchings, and every two Hamiltonian cycles, can be transformed into each other by a symmetry of the graph. There are 28 six-vertex cycles in the Heawood graph, each 6-cycle is disjoint from exactly three other 6-cycles, among these three 6-cycles, each one is the symmetric difference of the other two. The graph with one node per 6-cycle, and one edge for each disjoint pair of 6-cycles, is the Coxeter graph, the Heawood graph is a toroidal graph, that is, it can be embedded without crossings onto a torus. The graph is named after Percy John Heawood, who in 1890 proved that in every subdivision of the torus into polygons, the Heawood graph forms a subdivision of the torus with seven mutually adjacent regions, showing that this bound is tight. The Heawood graph is also the Levi graph of the Fano plane, with this interpretation, the 6-cycles in the Heawood graph correspond to triangles in the Fano plane. Also, the Heawood graph is the Bruhat-Tits building of the group SL3, the Heawood graph has crossing number 3, and is the smallest cubic graph with that crossing number. Including the Heawood graph, there are 8 distinct graphs of order 14 with crossing number 3, the automorphism group of the Heawood graph is isomorphic to the projective linear group PGL2, a group of order 336. It acts transitively on the vertices, on the edges and on the arcs of the graph, therefore the Heawood graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex, more strongly, the Heawood graph is 4-arc-transitive. According to the Foster census, the Heawood graph, referenced as F014A, is the cubic symmetric graph on 14 vertices. The characteristic polynomial of the Heawood graph is 6 and it is the only graph with this characteristic polynomial, making it a graph determined by its spectrum
27.
Pappus graph
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In the mathematical field of graph theory, the Pappus graph is a bipartite 3-regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek mathematician who is believed to have discovered the hexagon theorem describing the Pappus configuration, all the cubic distance-regular graphs are known, the Pappus graph is one of the 13 such graphs. The Pappus graph has crossing number 5, and is the smallest cubic graph with that crossing number. It has girth 6, diameter 4, radius 4, chromatic number 2, the Pappus graph has a chromatic polynomial equal to, x }. The first Pappus graph can be embedded in the torus to form a self-Petrie dual regular map with nine hexagonal faces, the two regular toroidal maps are dual to each other. The automorphism group of the Pappus graph is a group of order 216 and it acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Pappus graph is a symmetric graph and it has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Pappus graph, referenced as F018A, is the cubic symmetric graph on 18 vertices. The characteristic polynomial of the Pappus graph is x 46 and it is the only graph with this characteristic polynomial, making it a graph determined by its spectrum
28.
Dodecahedron
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In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does
29.
Desargues graph
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In the mathematical field of graph theory, the Desargues graph is a distance-transitive cubic graph with 20 vertices and 30 edges. The name Desargues graph has also used to refer to a ten-vertex graph, the complement of the Petersen graph. There are several different ways of constructing the Desargues graph, It is the generalized Petersen graph G, the Desargue graph consists of the 20 edges of these two polygons together with an additional 10 edges connecting points of one decagon to the corresponding points of the other. It is the Levi graph of the Desargues configuration and this configuration consists of ten points and ten lines describing two perspective triangles, their center of perspectivity, and their axis of perspectivity. The Desargues graph has one vertex for each point, one vertex for each line and it is the bipartite double cover of the Petersen graph, formed by replacing each Petersen graph vertex by a pair of vertices and each Petersen graph edge by a pair of crossed edges. It is the bipartite Kneser graph H5,2, equivalently, the Desargues graph is the induced subgraph of the 5-dimensional hypercube determined by the vertices of weight 2 and weight 3. The Desargues graph is Hamiltonian and can be constructed from the LCF notation,5, the Desargues graph is a symmetric graph, it has symmetries that take any vertex to any other vertex and any edge to any other edge. Its symmetry group has order 240, and is isomorphic to the product of a group on 5 points with a group of order 2. The symmetric group on five points is also the group of the Petersen graph. The generalized Petersen graph G is vertex-transitive if and only if n =10 and k =2 or if k2 ≡ ±1 and is only in the following seven cases. So the Desargues graph is one of only seven symmetric Generalized Petersen graphs, among these seven graphs are the cubical graph G, the Petersen graph G, the Möbius–Kantor graph G, the dodecahedral graph G and the Nauru graph G. The characteristic polynomial of the Desargues graph is 4554, therefore the Desargues graph is an integral graph, its spectrum consists entirely of integers. In chemistry, the Desargues graph is known as the Desargues–Levi graph, in this application, the thirty edges of the graph correspond to pseudorotations of the ligands. The Desargues graph has crossing number 6, and is the smallest cubic graph with that crossing number. It is the only known cubic partial cube. The Desargues graph has chromatic number 2, chromatic index 3, radius 5, diameter 5 and it is also a 3-vertex-connected and a 3-edge-connected Hamiltonian graph. All the cubic distance-regular graphs are known, the Desargues graph is one of the 13 such graphs. The Desargues graph can be embedded as a self-Petrie dual regular map in the manifold of genus 6
30.
Nauru graph
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In the mathematical field of graph theory, the Nauru graph is a symmetric bipartite cubic graph with 24 vertices and 36 edges. It was named by David Eppstein after the star in the flag of Nauru. It has chromatic number 2, chromatic index 3, diameter 4, radius 4 and it is also a 3-vertex-connected and 3-edge-connected graph. The Nauru graph requires at least eight crossings in any drawing of it in the plane and it is one of five non-isomorphic graphs tied for being the smallest cubic graph that requires eight crossings. Another of these five graphs is the McGee graph also known as the -cage, the Nauru graph is Hamiltonian and can be described by the LCF notation,4. There is also a construction of the Nauru graph. Take three distinguishable objects and place them in four boxes, no more than one object per box. There are 24 ways to so distribute the objects, corresponding to the 24 vertices of the graph, the resulting state-transition graph is the Nauru graph. The automorphism group of the Nauru graph is a group of order 144 and it is isomorphic to the direct product of the symmetric groups S4 and S3 and acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Nauru graph is a symmetric graph and it has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Nauru graph is the cubic symmetric graph on 24 vertices. The generalized Petersen graph G is vertex-transitive if and only if n =10 and k =2 or if k2 ≡ ±1 and is only in the following seven cases. So the Nauru graph is one of only seven symmetric Generalized Petersen graphs, among these seven graphs are the cubical graph G, the Petersen graph G, the Möbius–Kantor graph G, the dodecahedral graph G and the Desargues graph G. The Nauru graph is a Cayley graph of S4, the group of permutations on four elements. The characteristic polynomial of the Nauru graph is equal to 63 x 436, making it an integral graph—a graph whose spectrum consists entirely of integers. One of these two forms a torus, so the Nauru graph is a toroidal graph, it consists of 12 hexagonal faces together with the 24 vertices and 36 edges of the Nauru graph. The dual graph of this embedding is a symmetric 6-regular graph with 12 vertices and 36 edges, the other symmetric embedding of the Nauru graph has six dodecagonal faces, and forms a surface of genus 4. Its dual is not a graph, since each face shares three edges with four other faces, but a multigraph
31.
Generalized Petersen graph
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In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph, the generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and these graphs were given their name in 1969 by Mark Watkins. In Watkins notation, G is a graph with vertex set, any generalized Petersen graph can also be constructed as a voltage graph from a graph with two vertices, two self-loops, and one other edge. The Petersen graph itself is G or +, among the generalized Petersen graphs are the n-prism G the Dürer graph G, the Möbius-Kantor graph G, the dodecahedron G, the Desargues graph G and the Nauru graph G. Four generalized Petersen graphs – the 3-prism, the 5-prism, the Dürer graph, and G – are among the seven graphs that are cubic, 3-vertex-connected and this family of graphs possesses a number of interesting properties. For example, G is vertex-transitive if and only if n =10 and it is edge-transitive only in the following seven cases, =. These seven graphs are therefore the only symmetric generalized Petersen graphs and it is bipartite if and only if n is even and k is odd. It is a Cayley graph if and only if k2 ≡1 and it is hypohamiltonian when n is congruent to 5 modulo 6 and k is 2, n−2, /2, or /2. It is also non-Hamiltonian when n is divisible by four, at least equal to 8, in all other cases it has a Hamiltonian cycle. When n is congruent to 3 modulo 6 and k is 2, G has exactly three Hamiltonian cycles, for G, the number of Hamiltonian cycles can be computed by a formula that depends on the congruence class of n modulo six and involves the Fibonacci numbers. The Petersen graph itself is the only generalized Petersen graph that is not 3-edge-colorable, the generalized Petersen graph G is one of the few graphs known to have only one 3-edge-coloring. Every generalized Petersen graph is a unit distance graph
32.
F26A graph
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In the mathematical field of graph theory, the F26A graph is a symmetric bipartite cubic graph with 26 vertices and 39 edges. It has chromatic number 2, chromatic index 3, diameter 5, radius 5 and it is also a 3-vertex-connected and 3-edge-connected graph. The F26A graph is Hamiltonian and can be described by the LCF notation 13, the automorphism group of the F26A graph is a group of order 78. It acts transitively on the vertices, on the edges, therefore the F26A graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex, according to the Foster census, the F26A graph is the only cubic symmetric graph on 26 vertices. It is also a Cayley graph for the dihedral group D26, generated by a, ab, and ab4, where, D26 = ⟨ a, b | a 2 = b 13 =1, a b a = b −1 ⟩. The F26A graph is the smallest cubic graph where the group acts regularly on arcs. The characteristic polynomial of the F26A graph is equal to 6, the F26A graph can be embedded as a chiral regular map in the torus, with 13 hexagonal faces
33.
Coxeter graph
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In the mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. All the cubic distance-regular graphs are known, the Coxeter graph is one of the 13 such graphs. It is named after Harold Scott MacDonald Coxeter, the Coxeter graph has chromatic number 3, chromatic index 3, radius 4, diameter 4 and girth 7. It is also a 3-vertex-connected graph and a 3-edge-connected graph, the Coxeter graph is hypohamiltonian, it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It has rectilinear crossing number 11, and is the smallest cubic graph with crossing number currently known. The simplest construction of a Coxeter graph is from a Fano plane, take the 7C3 =35 possible 3-combinations on 7 objects. Discard the 7 triplets that correspond to the lines of the Fano plane, link two triplets if they are disjoint. The result is the Coxeter graph, the Coxeter graph may be derived from the Hoffman-Singleton graph. Take any vertex v in the Hoffman-Singleton graph, there is an independent set of size 15 that includes v. Delete the 7 neighbors of v, and the whole independent set including v, leaving behind the Coxeter graph. The automorphism group of the Coxeter graph is a group of order 336 and it acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Coxeter graph is a symmetric graph and it has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Coxeter graph, referenced as F28A, is the cubic symmetric graph on 28 vertices. The Coxeter graph is uniquely determined by its graph spectrum. The characteristic polynomial of the Coxeter graph is 876 and it is the only graph with this characteristic polynomial, making it a graph determined by its spectrum
34.
Dyck graph
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In the mathematical field of graph theory, the Dyck graph is a 3-regular graph with 32 vertices and 48 edges, named after Walther von Dyck. It is Hamiltonian with 120 distinct Hamiltonian cycles and it has chromatic number 2, chromatic index 3, radius 5, diameter 5 and girth 6. It is also a 3-vertex-connected and a 3-edge-connected graph, the Dyck graph is a toroidal graph, and the dual of its symmetric toroidal embedding is the Shrikhande graph, a strongly regular graph both symmetric and hamiltonian. The automorphism group of the Dyck graph is a group of order 192 and it acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Dyck graph is a symmetric graph and it has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Dyck graph, referenced as F32A, is the cubic symmetric graph on 32 vertices. The characteristic polynomial of the Dyck graph is equal to 996, the Dyck graph is the skeleton of a symmetric tessellation of a surface of genus three by twelve octagons, known as the Dyck map or Dyck tiling. The dual graph for this tiling is the complete tripartite graph K4,4,4
35.
Foster graph
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In the mathematical field of graph theory, the Foster graph is a bipartite 3-regular graph with 90 vertices and 135 edges. The Foster graph is Hamiltonian and has chromatic number 2, chromatic index 3, radius 8, diameter 8 and it is also a 3-vertex-connected and 3-edge-connected graph. All the cubic distance-regular graphs are known, the Foster graph is one of the 13 such graphs. It is the unique distance-transitive graph with intersection array and it can be constructed as the incidence graph of the partial linear space which is the unique triple cover with no 8-gons of the generalized quadrangle GQ. It is named after R. M. Foster, whose Foster census of cubic symmetric graphs included this graph, the automorphism group of the Foster graph is a group of order 4320. It acts transitively on the vertices, on the edges and on the arcs of the graph, therefore, the Foster graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex, according to the Foster census, the Foster graph, referenced as F90A, is the only cubic symmetric graph on 90 vertices. The characteristic polynomial of the Foster graph is equal to 918 x 1018912. Biggs, N. L. Boshier, A. G. Shawe-Taylor, J. Cubic distance-regular graphs, Journal of the London Mathematical Society,33, 385–394, doi,10. 1112/jlms/s2-33.3.385, van Dam, Edwin R. Haemers, Willem H. Spectral characterizations of some distance-regular graphs, Journal of Algebraic Combinatorics,15, 189–202, doi,10. 1023/A,1013847004932, MR1887234. Van Maldeghem, Hendrik, Ten exceptional geometries from trivalent distance regular graphs, Annals of Combinatorics,6, 209–228, doi,10. 1007/PL00012587, MR1955521
36.
Cycle graph
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In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called Cn, the number of vertices in Cn equals the number of edges, and every vertex has degree 2, that is, every vertex has exactly two edges incident with it. There are many synonyms for cycle graph and these include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or n-gon are also often used, a cycle with an even number of vertices is called an even cycle, a cycle with an odd number of vertices is called an odd cycle. A cycle graph is, 2-edge colorable, if and only if it has an number of vertices 2-regular 2-vertex colorable, if. More generally, a graph is bipartite if and only if it has no odd cycles, in particular, there exist symmetries taking any vertex to any other vertex, and any edge to any other edge, so the n-cycle is a symmetric graph. A directed cycle graph is a version of a cycle graph. In a directed graph, a set of edges which contains at least one edge from each directed cycle is called a feedback arc set, similarly, a set of vertices containing at least one vertex from each directed cycle is called a feedback vertex set. A directed cycle graph has uniform in-degree 1 and uniform out-degree 1, directed cycle graphs are Cayley graphs for cyclic groups. Complete bipartite graph Complete graph Null graph Path graph Weisstein, Eric W. Cycle Graph
37.
Hypercube graph
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In graph theory, the hypercube graph Qn is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cubical graph Q3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube, Qn has 2n vertices, 2n−1n edges, and is a regular graph with n edges touching each vertex. It is the n-fold Cartesian product of the complete graph. Hypercube graphs should not be confused with cubic graphs, which are graphs that have exactly three edges touching each vertex, the only hypercube graph Qn that is a cubic graph is the cubical graph Q3. Equivalently, it may be constructed using 2n vertices labeled with n-bit binary numbers, the joining edges form a perfect matching. Another construction of Qn is the Cartesian product of n two-vertex complete graphs K2, more generally the Cartesian product of copies of a complete graph is called a Hamming graph, the hypercube graphs are examples of Hamming graphs. The graph Q0 consists of a vertex, while Q1 is the complete graph on two vertices and Q2 is a cycle of length 4. The graph Q3 is the 1-skeleton of a cube, a cubical graph, the graph Q4 is the Levi graph of the Möbius configuration. It is also the knights graph for a toroidal 4 ×4 chessboard, every hypercube graph is bipartite, it can be colored with only two colors. Every hypercube Qn with n >1 has a Hamiltonian cycle, additionally, a Hamiltonian path exists between two vertices u and v if and only if they have different colors in a 2-coloring of the graph. Both facts are easy to prove using the principle of induction on the dimension of the hypercube, hamiltonicity of the hypercube is tightly related to the theory of Gray codes. More precisely there is a correspondence between the set of n-bit cyclic Gray codes and the set of Hamiltonian cycles in the hypercube Qn. An analogous property holds for acyclic n-bit Gray codes and Hamiltonian paths, a lesser known fact is that every perfect matching in the hypercube extends to a Hamiltonian cycle. The question whether every matching extends to a Hamiltonian cycle remains an open problem, the hypercube graph Qn, is the Hasse diagram of a finite Boolean algebra. is a median graph. Every median graph is a subgraph of a hypercube, and can be formed as a retraction of a hypercube. has more than 22n-2 perfect matchings. is arc transitive. The symmetries of graphs can be represented as signed permutations. Contains all the cycles of length 4,6, 2n and is thus a bipancyclic graph. is a n-vertex-connected graph, by Balinskis theorem is planar if and only if n ≤3. For larger values of n, the hypercube has genus 2n −3 +1. has exactly 22 n − n −1 ∏ k =2 n k spanning trees
38.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
39.
Icosahedron
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In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedra or icosahedrons. There are many kinds of icosahedra, with some being more symmetrical than others, the best known is the Platonic, convex regular icosahedron. There are two objects, one convex and one concave, that can both be called regular icosahedra, each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. The term regular icosahedron generally refers to the variety, while the nonconvex form is called a great icosahedron. Its dual polyhedron is the dodecahedron having three regular pentagonal faces around each vertex. The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra, like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges and its dual polyhedron is the great stellated dodecahedron, having three regular star pentagonal faces around each vertex. Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron and it is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron, of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them, other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are referred to as such. A regular icosahedron can be distorted or marked up as a lower symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron. This can be seen as a truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently, pyritohedral symmetry has the symbol, with order 24. Tetrahedral symmetry has the symbol, +, with order 12 and these lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles. These symmetries offer Coxeter diagrams, and respectively, each representing the lower symmetry to the regular icosahedron, the coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form. These coordinates represent the truncated octahedron with alternated vertices deleted and this construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector, where ϕ is the golden ratio
40.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2
41.
Icosidodecahedron
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In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly and its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, the icosidodecahedron can be considered a pentagonal gyrobirotunda, as a combination of two rotundae. In this form its symmetry is D5d, order 20, the wire-frame figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices. Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the permutations of. The icosidodecahedron has four special orthogonal projections, centered on a vertex, an edge, a face. The last two correspond to the A2 and H2 Coxeter planes, the icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The icosidodecahedron is a dodecahedron and also a rectified icosahedron. With orbifold notation symmetry of all of these tilings are wythoff construction within a fundamental domain of symmetry. The icosidodecahedron is related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotunda connected as mirror images, the icosidodecahedron can therefore be called a pentagonal gyrobirotunda with the gyration between top and bottom halves. Eight uniform star polyhedra share the same vertex arrangement, of these, two also share the same edge arrangement, the small icosihemidodecahedron, and the small dodecahemidodecahedron. The vertex arrangement is shared with the compounds of five octahedra. In four-dimensional geometry the icosidodecahedron appears in the regular 600-cell as the slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words, the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons, six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the icosidodecahedron, in the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids
42.
Rado graph
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Every finite or countably infinite graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds up the subgraph one vertex at a time. The Rado graph is symmetric, any isomorphism of its induced subgraphs can be extended to a symmetry of the whole graph. In model theory, the Rado graph forms an example of a model of an ω-categorical. The names of this graph honor Richard Rado, Paul Erdős and it appears even earlier in the work of Wilhelm Ackermann. The Rado graph was first constructed by Ackermann in two ways, with either the hereditarily finite sets or the natural numbers. Erdős & Rényi constructed the Rado graph as the graph on a countable number of points. They proved that it has infinitely many automorphisms, and their argument also shows that it is unique though they did not mention this explicitly. Richard Rado rediscovered the Rado graph as a graph. Ackermann and Rado constructed the Rado graph using the BIT predicate as follows and they identified the vertices of the graph with the natural numbers 0,1,2. An edge connects vertices x and y in the graph whenever the xth bit of the representation of y is nonzero. Thus, for instance, the neighbors of vertex 0 consist of all odd-numbered vertices, the neighbors of vertex 1 consist of vertex 0 and all vertices with numbers congruent to 2 or 3 modulo 4, every one of which is greater than 1. The Rado graph arises almost surely in the Erdős–Rényi model of a graph on countably many vertices. Specifically, one may form a graph by choosing, independently and with probability 1/2 for each pair of vertices. With probability 1 the resulting graph has the property, and is therefore isomorphic to the Rado graph. This construction also works if any fixed probability p not equal to 0 or 1 is used in place of 1/2 and this result, shown by Paul Erdős and Alfréd Rényi in 1963, justifies the definite article in the common alternative name “the random graph” for the Rado graph. For finite graphs, repeatedly drawing a graph from the Erdős–Rényi model will lead to different graphs. Since one obtains the same process by inverting all choices. A similar construction can be based on Skolems paradox, the fact there exists a countable model for the first-order theory of sets
43.
Connectivity (graph theory)
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It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network, a graph is connected when there is a path between every pair of vertices. In a connected graph, there are no unreachable vertices, a graph that is not connected is disconnected. A graph G is said to be disconnected if there exist two nodes in G such that no path in G has those nodes as endpoints, a graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected, in an undirected graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise, they are called disconnected. If the two vertices are connected by a path of length 1, i. e. by a single edge. A graph is said to be connected if every pair of vertices in the graph is connected, a connected component is a maximal connected subgraph of G. Each vertex belongs to exactly one connected component, as does each edge, a directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected graph. It is connected if it contains a path from u to v or a directed path from v to u for every pair of vertices u, v. It is strongly connected, diconnected, or simply strong if it contains a path from u to v. The strong components are the maximal strongly connected subgraphs, a cut, vertex cut, or separating set of a connected graph G is a set of vertices whose removal renders G disconnected. The connectivity or vertex connectivity κ is the size of a minimal vertex cut, a graph is called k-connected or k-vertex-connected if its vertex connectivity is k or greater. In particular, a graph with n vertices, denoted Kn, has no vertex cuts at all. A vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v. The local connectivity κ is the size of a smallest vertex cut separating u and v. Local connectivity is symmetric for undirected graphs, a graph G which is connected but not 2-connected is sometimes called separable. Analogous concepts can be defined for edges, in the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a bridge. More generally, a cut of G is a set of edges whose removal renders the graph disconnected. A graph is called k-edge-connected if its edge connectivity is k or greater, if u and v are vertices of a graph G, then a collection of paths between u and v is called independent if no two of them share a vertex