1.
Vertex (graph theory)
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In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices, a vertex w is said to be adjacent to another vertex v if the graph contains an edge. The neighborhood of a v is an induced subgraph of the graph. The degree of a vertex, denoted

2.
Edge (graph theory)
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This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges, for instance, α is the independence number of a graph, α′ is the matching number of the graph, which equals the independence number of its line graph. Similarly, χ is the number of a graph, χ ′ is the chromatic index of the graph. Achromatic The achromatic number of a graph is the number of colors in a complete coloring. A graph is acyclic if it has no cycles, an acyclic undirected graph is the same thing as a forest. Acyclic directed graphs are often called directed acyclic graphs. An acyclic coloring of a graph is a proper coloring in which every two color classes induce a forest. Adjacent The relation between two vertices that are both endpoints of the same edge, α For a graph G, α is its independence number, and α′ is its matching number. Alternating In a graph with a matching, a path is a path whose edges alternate between matched and unmatched edges. An alternating cycle is, similarly, a cycle whose edges alternate between matched and unmatched edges, an augmenting path is an alternating path that starts and ends at unsaturated vertices. A larger matching can be found as the difference of the matching and the augmenting path. Anti-edge Synonym for non-edge, a pair of non-adjacent vertices, anti-triangle A three-vertex independent set, the complement of a triangle. An apex graph is a graph in which one vertex can be removed, the removed vertex is called the apex. A k-apex graph is a graph that can be made planar by the removal of k vertices, Synonym for universal vertex, a vertex adjacent to all other vertices. Arborescence Synonym for a rooted and directed tree, see tree, arrow An ordered pair of vertices, such as an edge in a directed graph. An arrow has a x, a head y. The arrow is the arrow of the arrow. Articulation point A vertex in a graph whose removal would disconnect the graph

3.
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

4.
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

5.
Dihedral group
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3

6.
Graph coloring
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In graph theory, graph coloring is a special case of graph labeling, it is an assignment of labels traditionally called colors to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a such that no two adjacent vertices share the same color, this is called a vertex coloring. Vertex coloring is the point of the subject, and other coloring problems can be transformed into a vertex version. For example, a coloring of a graph is just a vertex coloring of its line graph. However, non-vertex coloring problems are often stated and studied as is and that is partly for perspective, and partly because some problems are best studied in non-vertex form, as for instance is edge coloring. The convention of using colors originates from coloring the countries of a map and this was generalized to coloring the faces of a graph embedded in the plane. By planar duality it became coloring the vertices, and in form it generalizes to all graphs. In mathematical and computer representations, it is typical to use the first few positive or nonnegative integers as the colors, in general, one can use any finite set as the color set. The nature of the coloring problem depends on the number of colors, graph coloring enjoys many practical applications as well as theoretical challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned and it has even reached popularity with the general public in the form of the popular number puzzle Sudoku. Graph coloring is still an active field of research. Note, Many terms used in this article are defined in Glossary of graph theory, the first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps. Guthrie’s brother passed on the question to his mathematics teacher Augustus de Morgan at University College, arthur Cayley raised the problem at a meeting of the London Mathematical Society in 1879. The same year, Alfred Kempe published a paper that claimed to establish the result, for his accomplishment Kempe was elected a Fellow of the Royal Society and later President of the London Mathematical Society. In 1890, Heawood pointed out that Kempe’s argument was wrong, however, in that paper he proved the five color theorem, saying that every planar map can be colored with no more than five colors, using ideas of Kempe. The proof went back to the ideas of Heawood and Kempe, the proof of the four color theorem is also noteworthy for being the first major computer-aided proof. Kempe had already drawn attention to the general, non-planar case in 1879, the conjecture remained unresolved for 40 years, until it was established as the celebrated strong perfect graph theorem by Chudnovsky, Robertson, Seymour, and Thomas in 2002. One of the applications of graph coloring, register allocation in compilers, was introduced in 1981

7.
Edge coloring
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In graph theory, an edge coloring of a graph is an assignment of colors to the edges of the graph so that no two adjacent edges have the same color. For example, the figure to the shows a edge coloring of a graph by the colors red, blue. Edge colorings are one of different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a graph using at most k different colors, for a given value of k. The minimum required number of colors for the edges of a graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizings theorem, the number of colors needed to color a simple graph is either its maximum degree Δ or Δ+1. For some graphs, such as graphs and high-degree planar graphs, the number of colors is always Δ, and for multigraphs. Many variations of the coloring problem, in which an assignments of colors to edges must satisfy other conditions than non-adjacency, have been studied. Edge colorings have applications in scheduling problems and in frequency assignment for fiber optic networks, a cycle graph may have its edges colored with two colors if the length of the cycle is even, simply alternate the two colors around the cycle. However, if the length is odd, three colors are needed, a complete graph Kn with n vertices is edge-colorable with n −1 colors when n is an even number, this is a special case of Baranyais theorem. Soifer provides the geometric construction of a coloring in this case, place n points at the vertices. For each color class, include one edge from the center to one of the polygon vertices, however, when n is odd, n colors are needed, each color can only be used for /2 edges, a 1/n fraction of the total. The case that n =3 gives the well-known Petersen graph, when n is 3,4, or 8, an edge coloring of On requires n +1 colors, but when it is 5,6, or 7, only n colors are needed. Here, two edges are considered to be adjacent when they share a common vertex, a proper edge coloring with k different colors is called a k-edge-coloring. A graph that can be assigned a k-edge-coloring is said to be k-edge-colorable, the smallest number of colors needed in a edge coloring of a graph G is the chromatic index, or edge chromatic number, χ′. The chromatic index is sometimes written using the notation χ1, in this notation. A graph is k-edge-chromatic if its chromatic index is exactly k, the chromatic index should not be confused with the chromatic number χ or χ0, the minimum number of colors needed in a proper vertex coloring of G

8.
Cage (graph theory)
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In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an -graph is defined to be a graph in which each vertex has exactly r neighbors and it is known that an -graph exists for any combination of r ≥2 and g ≥3. An -cage is an -graph with the fewest possible number of vertices, if a Moore graph exists with degree r and girth g, it must be a cage. Any -graph with exactly this many vertices is by definition a Moore graph, there may exist multiple cages for a given combination of r and g. For instance there are three nonisomorphic -cages, each with 70 vertices, the Balaban 10-cage, the Harries graph, but there is only one -cage, the Balaban 11-cage. A degree-one graph has no cycle, and a connected graph has girth equal to its number of vertices. The -cage is a complete graph Kr+1 on r+1 vertices, when r −1 is a prime power, the cages are the incidence graphs of projective planes. When r −1 is a power, the and cages are generalized polygons. Equivalently, g can be at most proportional to the logarithm of n, more precisely, g ≤2 log r −1 n + O. It is believed that this bound is tight or close to tight, the best known lower bounds on g are also logarithmic, but with a smaller constant factor. Specifically, the Ramanujan graphs satisfy the bound g ≥43 log r −1 n + O and it is unlikely that these graphs are themselves cages, but their existence gives an upper bound to the number of vertices needed in a cage. Biggs, Norman, Algebraic Graph Theory, Cambridge Mathematical Library, pp. 180–190, bollobás, Béla, Szemerédi, Endre, Girth of sparse graphs, Journal of Graph Theory,39, 194–200, doi,10. 1002/jgt.10023, MR1883596. Exoo, G, Jajcay, R, Dynamic Cage Survey, Dynamic Surveys, Electronic Journal of Combinatorics, erdős, Paul, Rényi, Alfréd, Sós, Vera T. On a problem of graph theory, Studia Sci, hartsfield, Nora, Ringel, Gerhard, Pearls in Graph Theory, A Comprehensive Introduction, Academic Press, pp. 77–81, ISBN 0-12-328552-6. The Petersen Graph, Cambridge University Press, pp. 183–213, lubotzky, A. Phillips, R. Sarnak, P. Ramanujan graphs, Combinatorica,8, 261–277, doi,10. 1007/BF02126799, MR963118. A family of graphs, Proc. Brouwer, Andries E. Cages Royle, Gordon, cubic Cages and Higher valency cages Weisstein, Eric W. Cage Graph

9.
Hamiltonian path
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In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle is a Hamiltonian path that is a cycle, determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete. Hamilton solved this problem using the calculus, an algebraic structure based on roots of unity with many similarities to the quaternions. This solution does not generalize to arbitrary graphs, in 18th century Europe, knights tours were published by Abraham de Moivre and Leonhard Euler. A Hamiltonian path or traceable path is a path that each vertex exactly once. A graph that contains a Hamiltonian path is called a traceable graph, a graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph, similar notions may be defined for directed graphs, where each edge of a path or cycle can only be traced in a single direction. A Hamiltonian decomposition is a decomposition of a graph into Hamiltonian circuits. All Hamiltonian graphs are biconnected, but a biconnected graph need not be Hamiltonian, an Eulerian graph G necessarily has an Euler tour, a closed walk passing through each edge of G exactly once. This tour corresponds to a Hamiltonian cycle in the line graph L, a tournament is Hamiltonian if and only if it is strongly connected. The number of different Hamiltonian cycles in an undirected graph on n vertices is. /2 and in a directed graph on n vertices is. These counts assume that cycles that are the same apart from their starting point are not counted separately, the best vertex degree characterization of Hamiltonian graphs was provided in 1972 by the Bondy–Chvátal theorem, which generalizes earlier results by G. A. Dirac and Øystein Ore. Both Diracs and Ores theorems can also be derived from Pósas theorem, hamiltonicity has been widely studied with relation to various parameters such as graph density, toughness, forbidden subgraphs and distance among other parameters. Dirac and Ores theorems basically state that a graph is Hamiltonian if it has enough edges, Bondy–Chvátal theorem A graph is Hamiltonian if and only if its closure is Hamiltonian. Ore A graph with n vertices is Hamiltonian if, for pair of non-adjacent vertices. Meyniel A strongly connected directed graph with n vertices is Hamiltonian if the sum of full degrees of every pair of distinct non-adjacent vertices is greater than or equal to 2n −1

10.
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

11.
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

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.
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

14.
Book thickness
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In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings into a book, a collection of half-planes all having the same line as their boundary. Usually, the vertices of the graph are required to lie on this line, called the spine. The book thickness of a graph is the smallest possible number of half-planes for any book embedding of the graph, book thickness is also called pagenumber, stacknumber or fixed outerthickness. Book embeddings have also used to define several other graph invariants including the pagewidth. Every graph with n vertices has book thickness at most ⌈ n /2 ⌉, the graphs with book thickness one are the outerplanar graphs. The graphs with book thickness at most two are the graphs, which are always planar, more generally, every planar graph has book thickness at most four. All minor-closed graph families, and in particular the graphs with bounded treewidth or bounded genus and it is NP-hard to determine the exact book thickness of a given graph, with or without knowing a fixed vertex ordering along the spine of the book. Book embedding also has applications in graph drawing, where two of the standard visualization styles for graphs, arc diagrams and circular layouts, can be constructed using book embeddings. A book embedding of this graph can be used to design a schedule that all the traffic move across the intersection with as few signal phases as possible. Other applications of book embeddings include abstract algebra and knot theory, there are several open problems concerning book thickness. It is unknown whether the book thickness of a graph can be bounded by a function of the book thickness of its subdivisions. And, although every planar graph has book thickness at most four, the notion of a book, as a topological space, was defined by C. A. Persinger and Gail Atneosen in the 1960s. As part of work, Atneosen already considered embeddings of graphs in books. In the early 1970s, Paul C, Kainen and L. Taylor Ollman developed a more restricted type of embedding that came to be used in most subsequent research. In their formulation, the vertices must be placed along the spine of the book. A book is a kind of topological space, also called a fan of half-planes. The k-page book crossing number of G is the number of crossings in a k-page book drawing. A book embedding of G onto B is a drawing that forms a graph embedding of G into B

15.
Queue number
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In mathematics, the queue number of a graph is a graph invariant defined analogously to stack number using first-in first-out orderings in place of last-in first-out orderings. A queue layout of a graph is defined by a total ordering of the vertices of the graph together with a partition of the edges into a number of queues. The queue number qn of a graph G is the number of queues in a queue layout. The nesting condition ensures that, when a vertex is reached, edges that are assigned to the same queue are not allowed to cross each other, but crossings are allowed between edges that belong to different queues. Queue layouts were defined by Heath & Rosenberg, by analogy to previous work on book embeddings of graphs, every tree has queue number 1, with a vertex ordering given by a breadth-first traversal. Pseudoforests and grid graphs also have queue number 1, outerplanar graphs have queue number at most 2, the 3-sun graph is an example of an outerplanar graph whose queue number is exactly 2. Series-parallel graphs have queue number at most 3, binary de Bruijn graphs have queue number 2. The d-dimensional hypercube graph has queue number at most d −1, the queue numbers of complete graphs Kn and complete bipartite graphs Ka, b are known exactly, they are ⌊ n /2 ⌋ and min respectively. Conversely, every arched leveled planar graph has a 1-queue layout, Heath, Leighton & Rosenberg conjectured that every planar graph has bounded queue number, but this remains unsolved. If the queue number of graphs is bounded, then the same is true for 1-planar graphs. Graphs with low queue number are sparse graphs, 1-queue graphs with n vertices have at most 2n −3 edges, and more generally graphs with queue number q have at most 2qn − q edges. This implies that these also have small chromatic number, in particular 1-queue graphs are 3-colorable. In the other direction, a bound on the number of edges implies a weaker bound on the queue number, graphs with n vertices. This bound is close to tight, because for random d-regular graphs the number is, with high probability. Graphs with queue number 1 have book thickness at most 2, for any fixed vertex ordering, the product of the book thickness and queue numbers for that ordering is at least as large as the cutwidth of the graph divided by its maximum degree. The book thickness may be larger than the queue number, ternary Hamming graphs have logarithmic queue number. It remains unknown whether the book thickness can be bounded by any function of the queue number, Heath, Leighton & Rosenberg conjectured that the queue number is at most a linear function of the book thickness, but no functional bound in this direction is known either. It is known that, if all bipartite graphs with 3-page book embeddings have bounded queue number, however, the queue number was subsequently shown to be bounded by a function of the treewidth

16.
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

17.
Dodecagon
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In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with sides of the same length. It has twelve lines of symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a hexagon, t, or a twice-truncated triangle. The internal angle at each vertex of a regular dodecagon is 150°, as 12 =22 ×3, regular dodecagon is constructible using compass and straightedge, Coxeter states that every parallel-sided 2m-gon can be divided into m/2 rhombs. For the dodecagon, m=6, and it can be divided into 15 rhombs and this decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons, the regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries, each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges, the interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes, a regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a antiprism with the same D5d, symmetry. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensionare the 24-cell, snub 24-cell, 6-6 duoprism, in 6 dimensions 6-cube, 6-orthoplex,221,122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell, a dodecagram is a 12-sided star polygon, represented by symbol. There is one regular star polygon, using the same vertices, but connecting every fifth point. There are also three compounds, is reduced to 2 as two hexagons, and is reduced to 3 as three squares, is reduced to 4 as four triangles, and is reduced to 6 as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams can produce intermediate star polygon forms with equal spaced vertices. A truncated hexagon is a dodecagon, t=, a quasitruncated hexagon, inverted as, is a dodecagram, t=