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 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
5.
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
6.
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
7.
Tree (graph theory)
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In mathematics, and more specifically in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, any connected graph is a tree. A forest is a disjoint union of trees, the various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree itself has been defined by authors as a directed graph. The term tree was coined in 1857 by the British mathematician Arthur Cayley, a tree is an undirected graph G that satisfies any of the following equivalent conditions, G is connected and has no cycles. Any two vertices in G can be connected by a simple path. If G has finitely many vertices, say n of them, then the statements are also equivalent to any of the following conditions. G has no simple cycles and has n −1 edges, an internal vertex is a vertex of degree at least 2. Similarly, a vertex is a vertex of degree 1. An irreducible tree is a tree in which there is no vertex of degree 2, a forest is an undirected graph, all of whose connected components are trees, in other words, the graph consists of a disjoint union of trees. Equivalently, a forest is an acyclic graph. As special cases, an empty graph, a tree. A polytree is an acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, a directed tree is a directed graph which would be a tree if the directions on the edges were ignored, i. e. a polytree. Some authors restrict the phrase to the case where the edges are all directed towards a particular vertex, a rooted tree is a tree in which one vertex has been designated the root. The edges of a tree can be assigned a natural orientation, either away from or towards the root. The tree-order is the ordering on the vertices of a tree with u < v if. A rooted tree which is a subgraph of some graph G is a tree if the ends of every edge in G are comparable in this tree-order whenever those ends are vertices of the tree
8.
Unit distance graph
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Edges of unit distance graphs sometimes cross each other, so they are not always planar, a unit distance graph without crossings is called a matchstick graph. The Hadwiger–Nelson problem concerns the number of unit distance graphs. It is known that there exist unit distance graphs requiring four colors in any coloring. Another important open problem concerning unit distance graphs asks how many edges they can have relative to their number of vertices, however, the same is not true for other commonly-used graph products. For instance, the Möbius-Kantor graph has a drawing of this type, according to this looser definition of a unit distance graph, all generalized Petersen graphs are unit distance graphs. In order to distinguish the two definitions, the graphs in which non-edges are required to be a distance apart may be called strict unit distance graphs. Paul Erdős posed the problem of estimating how many pairs of points in a set of n points could be at unit distance from each other, in graph theoretic terms, how dense can a unit distance graph be. The hypercube graph provides a bound on the number of unit distances proportional to n log n. The Hamming graph meets this bound, with 27 vertices and 81 edges, for every algebraic number A, it is possible to find a unit distance graph G in which some pair of vertices are at distance A in all unit distance representations of G. The definition of a unit distance graph may naturally be generalized to any higher-dimensional Euclidean space. It is NP-hard, and more complete for the existential theory of the reals, to test whether a given graph is a unit distance graph. It is also NP-complete to determine whether a unit distance graph has a Hamiltonian cycle, unit disk graph, a graph on the plane that has an edge whenever two points are at distance at most one Beckman, F. S. Quarles, D. A. Jr. On isometries of Euclidean spaces, Proceedings of the American Mathematical Society,4, 810–815, doi,10. 2307/2032415, Erdős, Paul, On sets of distances of n points, American Mathematical Monthly,53, 248–250, doi,10. 2307/2305092, JSTOR2305092. Erdős, Paul, Simonovits, Miklós, On the chromatic number of graphs, Ars Combinatoria,9. Eleven unit distance embeddings of the Heawood graph, arXiv,0912.5395. Gervacio, Severino V. Lim, Yvette F. Maehara, Hiroshi, Planar unit-distance graphs having planar unit-distance complement, Discrete Mathematics,308, 1973–1984, doi,10. 1016/j. disc.2007.04.050. Horvat, Boris, Pisanski, Tomaž, Products of unit distance graphs, Discrete Mathematics,310, 1783–1792, doi,10. 1016/j. disc.2009.11.035, MR2610282. Itai, Alon, Papadimitriou, Christos H. Szwarcfiter, Jayme Luiz, Hamilton paths in graphs, SIAM Journal on Computing,11, 676–686, doi,10. 1137/0211056
9.
Bipartite graph
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In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the parts of the graph, equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. One often writes G = to denote a graph whose partition has the parts U and V. If | U | = | V |, that is, if all vertices on the same side of the bipartition have the same degree, then G is called biregular. When modelling relations between two different classes of objects, bipartite graphs very often arise naturally, a third example is in the academic field of numismatics. Ancient coins are made using two positive impressions of the design, the charts numismatists produce to represent the production of coins are bipartite graphs. More abstract examples include the following, Every tree is bipartite, cycle graphs with an even number of vertices are bipartite. Every planar graph whose faces all have even length is bipartite, special cases of this are grid graphs and squaregraphs, in which every inner face consists of 4 edges and every inner vertex has four or more neighbors. It follows that Km, n has mn edges, closely related to the complete bipartite graphs are the crown graphs, formed from complete bipartite graphs by removing the edges of a perfect matching. Hypercube graphs, partial cubes, and median graphs are bipartite, in these graphs, the vertices may be labeled by bitvectors, in such a way that two vertices are adjacent if and only if the corresponding bitvectors differ in a single position. A bipartition may be formed by separating the vertices whose bitvectors have a number of ones from the vertices with an odd number of ones. Trees and squaregraphs form examples of graphs, and every median graph is a partial cube. Bipartite graphs may be characterized in different ways, A graph is bipartite if. A graph is bipartite if and only if it is 2-colorable, the spectrum of a graph is symmetric if and only if its a bipartite graph. In bipartite graphs, the size of minimum vertex cover is equal to the size of the maximum matching, an alternative and equivalent form of this theorem is that the size of the maximum independent set plus the size of the maximum matching is equal to the number of vertices. In any graph without isolated vertices the size of the edge cover plus the size of a maximum matching equals the number of vertices. Perfection of bipartite graphs is easy to see but perfection of the complements of bipartite graphs is less trivial and this was one of the results that motivated the initial definition of perfect graphs. The bipartite graphs, line graphs of graphs, and their complements form four out of the five basic classes of perfect graphs used in the proof of the strong perfect graph theorem
10.
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
11.
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
12.
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
13.
Edge-graceful labeling
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In graph theory, an edge-graceful graph labeling is a type of graph labeling. This is a labeling for simple graphs in which no two distinct edges connect the two distinct vertices, no edge connects a vertex to itself, and the graph is connected. Edge-graceful labelings were first introduced by S. Lo in his seminal paper, given a graph G, we denote the set of edges by E and the vertices by V. Let q be the cardinality of E and p be that of V, the problem is to find a labeling for the edges such that all the labels from 1 to q are used once and the induced labels on the vertices run from 0 to p −1. In other words, the set for labels of the edges should be. A graph G is said to be if it admits an edge-graceful labeling. Consider a path with two vertices, P2, here the only possibility is to label the only edge in the graph 1. The induced labeling on the two vertices are both 1, appending an edge and a vertex to P2 gives P3, the path with three vertices. Denote the vertices by v1, v2, and v3, label the two edges in the following way, the edge is labeled 1 and labeled 2. The induced labelings on v1, v2, and v3 are then 1,0 and this is an edge-graceful labeling and so P3 is edge-graceful. Similarly, one can check that P4 is not edge-graceful, in general, Pm is edge-graceful when m is odd and not edge-graceful when it is even. This follows from a condition for edge-gracefulness. Consider the cycle with three vertices, C3, one can label the edges 1,2, and 3, and check directly that, along with the induced labeling on the vertices, this gives an edge-graceful labeling. Similar to paths, C m is edge-graceful when m is odd, an edge-graceful labeling of C5 is shown in the following figure, Lo gave a necessary condition for a graph to be edge-graceful. It is that a graph with q edges and p vertices is edge graceful only if q is congruent to p 2 modulo p. or, in symbols and this is referred to as Los condition in the literature. This follows from the fact that the sum of the labels of the vertices is twice the sum of the edges and this is useful for disproving a graph is edge-graceful. For instance, one can apply directly to the path. The Petersen graph is not edge-graceful, the star graph S m is edge-graceful when m is even and not when m is odd
14.
Isotoxal figure
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In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. The term isotoxal is derived from the Greek τοξον meaning arc, an isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons, in general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is a polygon with D2 symmetry. All regular polygons are isotoxal, having double the symmetry order. A regular 2n-gon is a polygon and can be marked with alternately colored vertices. An isotoxal polyhedron or tiling must be either isogonal or isohedral or both, regular polyhedra are isohedral, isogonal and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral, their duals are isohedral and isotoxal, not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. An isotoxal polyhedron has the dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids,8 formed by the Kepler–Poinsot polyhedra, cS1 maint, Multiple names, authors list Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Uniform polyhedra, Philosophical Transactions of the Royal Society of London, mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446
15.
Matchstick graph
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That is, it is a graph that has an embedding which is simultaneously a unit distance graph and a plane graph. Informally, matchstick graphs can be made by placing noncrossing matchsticks on a surface, hence the name. Much of the research on matchstick graphs has concerned regular graphs and this number is called the degree of the graph. It is known there are matchstick graphs that are regular of any degree up to 4. The complete graphs with one, two, and three vertices are all matchstick graphs and are 0-, 1-, and 2-regular respectively, in 1986, Heiko Harborth presented the graph that would bear his name, the Harborth Graph. With 104 edges and 52 vertices, is the smallest known example of a 4-regular matchstick graph and it is not possible for a regular matchstick graph to have degree greater than four. The smallest 3-regular matchstick graph without triangles has 20 vertices, as proved by Kurz, furthermore, they exhibit the smallest known example of a 3-regular matchstick graph of girth 5. It is NP-hard to test whether a given undirected graph can be realized as a matchstick graph. More precisely, this problem is complete for the theory of the reals. Kurz provides some easily tested necessary criteria for a graph to be a graph, but these are not also sufficient criteria. The numbers of distinct matchstick graphs are known for 1,2,3, uniformity of edge lengths has long been seen as a desirable quality in graph drawing, and some specific classes of planar graphs can always be drawn with completely uniform edges. Every tree can be drawn in such a way that, if the edges of the tree were replaced by infinite rays. For such a drawing, if the lengths of each edge are changed arbitrarily, without changing the slope of the edge, in particular, it is possible to choose all edges to have equal length, resulting in a realization of an arbitrary tree as a matchstick graph. In particular it is possible to normalize the edges so that all have the same length. Every matchstick graph is a unit distance graph, penny graphs are the graphs that can be represented by tangencies of non-overlapping unit circles. Every penny graph is a graph, but not vice versa
16.
Degree (graph theory)
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In graph theory, the degree of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice. The degree of a v is denoted deg or deg v. The maximum degree of a graph G, denoted by Δ, in the graph on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, all degrees are the same, the degree sum formula states that, given a graph G =, ∑ v ∈ V deg =2 | E |. The formula implies that in any graph, the number of vertices with odd degree is even and this statement is known as the handshaking lemma. The latter name comes from a mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even. The degree sequence of a graph is the non-increasing sequence of its vertex degrees. The degree sequence is a graph invariant so isomorphic graphs have the degree sequence. However, the sequence does not, in general, uniquely identify a graph, in some cases. The degree sequence problem is the problem of finding some or all graphs with the sequence being a given non-increasing sequence of positive integers. A sequence which is the sequence of some graph, i. e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the sum formula, any sequence with an odd sum, such as. The converse is true, if a sequence has an even sum. The construction of such a graph is straightforward, connect vertices with odd degrees in pairs by a matching, the question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can either be solved by the Erdős–Gallai theorem or the Havel–Hakimi algorithm, the problem of finding or estimating the number of graphs with a given degree sequence is a problem from the field of graph enumeration. A vertex with degree 0 is called an isolated vertex, a vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, is a pendant edge and this terminology is common in the study of trees in graph theory and especially trees as data structures. A vertex with degree n −1 in a graph on n vertices is called a dominating vertex, if each vertex of the graph has the same degree k the graph is called a k-regular graph and the graph itself is said to have degree k
17.
Claw-free graph
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In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph K1,3, a claw-free graph is a graph in which no induced subgraph is a claw, i. e. any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph. They are the subject of hundreds of research papers and several surveys. The line graph L of any graph G is claw-free, L has a vertex for every edge of G, line graphs may be characterized in terms of nine forbidden subgraphs, the claw is the simplest of these nine graphs. This characterization provided the motivation for studying claw-free graphs. The de Bruijn graphs are claw-free, one way to show this is via the construction of the de Bruijn graph for n-bit strings as the line graph of the de Bruijn graph for -bit strings. The complement of any graph is claw-free. These graphs include as a special case any complete graph, the same is true more generally for proper circular-arc graphs. The Moser spindle, a graph used to provide a lower bound for the chromatic number of the plane, is claw-free. The Schläfli graph, a regular graph with 27 vertices, is claw-free. It is straightforward to verify that a graph with n vertices and m edges is claw-free in time O. Therefore, using the Coppersmith–Winograd algorithm, the time for this claw-free recognition algorithm would be O. The worst case for this algorithm occurs when Ω vertices have Ω neighbors each, and the remaining vertices have few neighbors, so its total time is O = O. The numbers of connected claw-free graphs on n nodes, for n =1,2. are 1,1,2,5,14,50,191,881,4494,26389,184749. If the graphs are allowed to be disconnected, the numbers of graphs are even larger, a technique of Palmer, Read & Robinson allows the number of claw-free cubic graphs to be counted very efficiently, unusually for graph enumeration problems. Sumner and, independently, Las Vergnas proved that every connected graph with an even number of vertices has a perfect matching. That is, there exists a set of edges in the graph such that each vertex is an endpoint of one of the matched edges
18.
Induced subgraph
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In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset. Formally, let G = be any graph, and let S ⊂ V be any subset of vertices of G. Then the induced subgraph G is the graph whose vertex set is S, the same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph G may also be called the subgraph induced in G by S, important types of induced subgraphs include the following. Induced paths are induced subgraphs that are paths, conversely, in distance-hereditary graphs, every induced path is a shortest path. Induced cycles are induced subgraphs or cycles, the girth of a graph is defined by the length of its shortest cycle, which is always an induced cycle. According to the perfect graph theorem, induced cycles and their complements play a critical role in the characterization of perfect graphs. Cliques and independent sets are induced subgraphs that are complete graphs or edgeless graphs. The neighborhood of a vertex is the subgraph of all vertices adjacent to it. The induced subgraph isomorphism problem is a form of the isomorphism problem in which the goal is to test whether one graph can be found as an induced subgraph of another. Because it includes the problem as a special case, it is NP-complete
19.
Line graph
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In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L that represents the adjacencies between edges of G. The name line graph comes from a paper by Harary & Norman although both Whitney and Krausz used the construction before this, hassler Whitney proved that with one exceptional case the structure of a connected graph G can be recovered completely from its line graph. Line graphs are claw-free, and the graphs of bipartite graphs are perfect. Line graphs can be characterized by nine forbidden subgraphs, and can be recognized in linear time. Various generalizations of graphs have also been studied, including the line graphs of line graphs, line graphs of multigraphs, line graphs of hypergraphs. That is, it is the graph of the edges of G. The following figures show a graph and its line graph, each vertex of the line graph is shown labeled with the pair of endpoints of the corresponding edge in the original graph. For instance, the vertex on the right labeled 1,3 corresponds to the edge on the left between the blue vertices 1 and 3. Green vertex 1,3 is adjacent to three other vertices,1,4 and 1,2 and 4,3. Properties of a graph G that depend only on adjacency between edges may be translated into equivalent properties in L that depend on adjacency between vertices. For instance, a matching in G is a set of no two of which are adjacent, and corresponds to a set of vertices in L no two of which are adjacent, that is, an independent set. Thus, The line graph of a graph is connected. A line graph has a point if and only if the underlying graph has a bridge for which neither endpoint has degree one. A maximum independent set in a line graph corresponds to maximum matching in the original graph, the edge chromatic number of a graph G is equal to the vertex chromatic number of its line graph L. The line graph of a graph is vertex-transitive. If a graph G has an Euler cycle, that is, if G is connected and has an number of edges at each vertex. If two simple graphs are isomorphic then their line graphs are also isomorphic, the Whitney graph isomorphism theorem provides a converse to this for every but one graph. In the context of network theory, the line graph of a random network preserves many of the properties of the network such as the small-world property
20.
Graph isomorphism
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This kind of bijection is commonly described as edge-preserving bijection, in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs are called isomorphic and denoted as G ≃ H, a set of graphs isomorphic to each other is called an isomorphism class of graphs. The two graphs shown below are isomorphic, despite their different looking drawings, in the above definition, graphs are understood to be undirected non-labeled non-weighted graphs. with the following exception. For labeled graphs, two definitions of isomorphism are in use, under one definition, an isomorphism is a vertex bijection which is both edge-preserving and label-preserving. For example, the K2 graph with the two vertices labelled with 1 and 2 has a single automorphism under the first definition, in such cases two labeled graphs are sometimes said to be isomorphic if the corresponding underlying unlabeled graphs are isomorphic. For example, if a graph has exactly one cycle, then all graphs in its isomorphism class also have exactly one cycle, on the other hand, in the common case when the vertices of a graph are the integers 1,2. N, then the expression ∑ v ∈ V v ⋅ deg v may be different for two isomorphic graphs, the Whitney graph isomorphism theorem, shown by H. The Whitney graph theorem can be extended to hypergraphs, while graph isomorphism may be studied in a classical mathematical way, as exemplified by the Whitney theorem, it is recognized that it is a problem to be tackled with an algorithmic approach. The computational problem of determining whether two finite graphs are isomorphic is called the isomorphism problem. Its practical applications include primarily cheminformatics, mathematical chemistry, and electronic design automation, the graph isomorphism problem is one of few standard problems in computational complexity theory belonging to NP, but not known to belong to either of its well-known subsets, P and NP-complete. It is one of two, out of 12 total, problems listed in Garey & Johnson whose complexity remains unresolved. It is however known that if the problem is NP-complete then the hierarchy collapses to a finite level. In November 2015, László Babai, a mathematician and computer scientist at the University of Chicago and this work has not yet been vetted. In January 2017, Babai shortly retracted the quasi-polynomiality claim and stated a sub-exponential time time complexity bound instead and he restored the original claim five days later. Its generalization, the isomorphism problem, is known to be NP-complete. Graph homomorphism Graph automorphism problem Graph canonization Garey, Michael R. Johnson, computers and Intractability, A Guide to the Theory of NP-Completeness, W. H. Freeman, ISBN 0-7167-1045-5
21.
Graph invariant
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In other words, it is a property of the graph itself, not of a specific drawing or representation of the graph. Informally, the term graph invariant is used for properties expressed quantitatively, for example, the statement graph does not have vertices of degree 1 is a property while the number of vertices of degree 1 in a graph is an invariant. More formally, a property is a class of graphs with the property that any two isomorphic graphs either both belong to the class, or both do not belong to it. For instance, being a graph or being a chordal graph are hereditary properties. A graph property is monotone if every subgraph of a graph with property P also has property P, for instance, being a bipartite graph or being a triangle-free graph is monotone. Every monotone property is hereditary, but not necessarily vice versa, for instance, subgraphs of chordal graphs are not necessarily chordal, a graph property is minor-closed if every graph minor of a graph with property P also has property P. For instance, being a graph is minor-closed. Every minor-closed property is monotone, but not necessarily vice versa, for instance, for instance, the number of vertices is additive. A graph invariant is multiplicative if, for all two graphs G and H, the value of the invariant on the disjoint union of G and H is the product of the values on G, for instance, the Hosoya index is multiplicative. A graph invariant is maxing if, for all two graphs G and H, the value of the invariant on the disjoint union of G and H is the maximum of the values on G, for instance, the chromatic number is maxing. In addition, graph properties can be classified according to the type of graph they describe, whether the graph is undirected or directed, whether the property applies to multigraphs, etc. The target set of a function defines a graph invariant may be one of, A truth value, true or false. An integer, such as the number of vertices or chromatic number of a graph, a real number, such as the fractional chromatic number of a graph. A sequence of integers, such as the sequence of a graph. A polynomial, such as the Tutte polynomial of a graph, two graphs with the same invariants may or may not be isomorphic, however. A graph invariant I is called if the identity of the invariants I and I implies the isomorphism of the graphs G and H. Finding such an invariant would imply an easy solution to the graph isomorphism problem. However, even polynomial-valued invariants such as the polynomial are not usually complete
22.
Arboricity
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The arboricity of an undirected graph is the minimum number of forests into which its edges can be partitioned. Equivalently it is the number of spanning forests needed to cover all the edges of the graph. The figure shows the complete bipartite graph K4,4, with the colors indicating a partition of its edges into three forests, therefore, the arboricity of K4,4 is three. The arboricity of a graph is a measure of how dense the graph is, graphs with many edges have high arboricity, and graphs with high arboricity must have a dense subgraph. In more detail, as any n-vertex forest has at most n-1 edges, any planar graph with n vertices has at most 3 n −6 edges, from which it follows by Nash-Williams formula that planar graphs have arboricity at most three. Schnyder used a special decomposition of a graph into three forests called a Schnyder wood to find a straight-line embedding of any planar graph into a grid of small area. As a consequence, the arboricity can be calculated by a polynomial-time algorithm, the star arboricity of a graph is the size of the minimum forest, each tree of which is a star, into which the edges of the graph can be partitioned. If a tree is not a itself, its star arboricity is two, as can be seen by partitioning the edges into two subsets at odd and even distances from the tree root respectively. Therefore, the arboricity of any graph is at least equal to the arboricity. The linear arboricity of a graph is the number of linear forests into which the edges of the graph can be partitioned. The linear arboricity of a graph is related to its maximum degree. The pseudoarboricity of a graph is the number of pseudoforests into which its edges can be partitioned. Equivalently, it is the ratio of edges to vertices in any subgraph of the graph. As with the arboricity, the pseudoarboricity has a structure allowing it to be computed efficiently. The thickness of a graph is the number of planar subgraphs into which its edges can be partitioned. As any planar graph has arboricity three, the thickness of any graph is at least equal to a third of the arboricity, the degeneracy of a graph is the maximum, over all induced subgraphs of the graph, of the minimum degree of a vertex in the subgraph. The degeneracy of a graph with arboricity a is at least equal to a, the coloring number of a graph, also known as its Szekeres-Wilf number is always equal to its degeneracy plus 1. The strength of a graph is a value whose integer part gives the maximum number of disjoint spanning trees that can be drawn in a graph
23.
Star coloring
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In graph-theoretic mathematics, a star coloring of a graph G is a vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by Grünbaum, the star chromatic number χ s of G is the least number of colors needed to star color G. If we denote the chromatic number of a graph G by χ a, we have that χ a ≤ χ s. The star chromatic number has been proved to be bounded on every proper minor closed class by Nešetřil & Ossona de Mendez and this results was further generalized by Nešetřil & Ossona de Mendez to all low-tree-depth colorings. It was demonstrated by Albertson et al. that it is NP-complete to determine whether χ s ≤3, coleman & Moré showed that finding an optimal star coloring is NP-hard even when G is a bipartite graph. Albertson, Michael O. Chappell, Glenn G. Kierstead, Hal A. Kündgen, André, Ramamurthi, Radhika, Coloring with no 2-Colored P4s, The Electronic Journal of Combinatorics,11, MR2056078. Coleman, Thomas F. Moré, Jorge, Estimation of sparse Hessian matrices and graph coloring problems, Mathematical Programming,28, 243–270, doi,10. 1007/BF02612334, MR0736293. Fertin, Guillaume, Raspaud, André, Reed, Bruce, Star coloring of graphs, Journal of Graph Theory,47, 163–182, doi,10. 1002/jgt.20029, MR2089462. Grünbaum, Branko, Acyclic colorings of graphs, Israel Journal of Mathematics,14, 390–408, doi,10. 1007/BF02764716. Nešetřil, Jaroslav, Ossona de Mendez, Patrice, Tree depth, subgraph coloring and homomorphism bounds, European Journal of Combinatorics,27, 1022–1041, doi,10. 1016/j. ejc.2005.01.010, MR2226435. Star colorings and acyclic colorings, present at the Research Experiences for Graduate Students at the University of Illinois,2008
24.
Branch-decomposition
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In graph theory, a branch-decomposition of an undirected graph G is a hierarchical clustering of the edges of G, represented by an unrooted binary tree T with the edges of G as its leaves. Removing any edge from T partitions the edges of G into two subgraphs, and the width of the decomposition is the number of shared vertices of any pair of subgraphs formed in this way. And as with treewidth, many optimization problems may be solved efficiently for graphs of small branchwidth. However, unlike treewidth, the branchwidth of planar graphs may be computed exactly, branch-decompositions and branchwidth may also be generalized from graphs to matroids. An unrooted binary tree is an undirected graph with no cycles in which each non-leaf node has exactly three neighbors. A branch-decomposition may be represented by a binary tree T. If e is any edge of the tree T, then removing e from T partitions it into two subtrees T1 and T2 and this partition of T into subtrees induces a partition of the edges associated with the leaves of T into two subgraphs G1 and G2 of G. This partition of G into two subgraphs is called an e-separation, the width of the branch-decomposition is the maximum width of any of its e-separations. Carving width is a concept defined similarly to branch width, except with edges replaced by vertices, branch width algorithms typically work by reducing to an equivalent carving width problem. In particular, the width of the medial graph of a planar graph is exactly twice the branch width of the original graph. It is NP-complete to determine whether a graph G has a branch-decomposition of width at most k, for planar graphs, the branchwidth can be computed exactly in polynomial time. This in contrast to treewidth for which the complexity on planar graphs is a well known open problem and this was later sped up to O. It is also possible to define a notion of branch-decomposition for matroids that generalizes branch-decompositions of graphs, a branch-decomposition of a matroid is a hierarchical clustering of the matroid elements, represented as an unrooted binary tree with the elements of the matroid at its leaves. An e-separation may be defined in the way as for graphs. However, for graphs that are not trees, the branchwidth of the graph is equal to the branchwidth of its associated graphic matroid. The branchwidth of a matroid is equal to the branchwidth of its dual matroid, for any fixed constant k, the matroids with branchwidth at most k can be recognized in polynomial time by an algorithm that has access to the matroid via an independence oracle. By the Robertson–Seymour theorem, the graphs of branchwidth k can be characterized by a set of forbidden minors. The graphs of branchwidth 0 are the matchings, the forbidden minors are a two-edge path graph
25.
Metric space
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In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set, a metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is 3-dimensional Euclidean space, in fact, a metric is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the line segment connecting them. Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, since for any x, y ∈ M, The function d is also called distance function or simply distance. Often, d is omitted and one just writes M for a space if it is clear from the context what metric is used. Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations, to be a metric there shouldnt be any one-way roads. The triangle inequality expresses the fact that detours arent shortcuts, many of the examples below can be seen as concrete versions of this general idea. The real numbers with the function d = | y − x | given by the absolute difference. The rational numbers with the distance function also form a metric space. The positive real numbers with distance function d = | log | is a metric space. Any normed vector space is a space by defining d = ∥ y − x ∥. Examples, The Manhattan norm gives rise to the Manhattan distance, the maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from x to y. The British Rail metric on a vector space is given by d = ∥ x ∥ + ∥ y ∥ for distinct points x and y. The name alludes to the tendency of railway journeys to proceed via London irrespective of their final destination, If is a metric space and X is a subset of M, then becomes a metric space by restricting the domain of d to X × X. The discrete metric, where d =0 if x = y and d =1 otherwise, is a simple but important example and this, in particular, shows that for any set, there is always a metric space associated to it. Using this metric, any point is a ball, and therefore every subset is open. A finite metric space is a metric space having a number of points
26.
Isometry
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In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective. Isometries are often used in constructions where one space is embedded in another space, for instance, the completion of a metric space M involves an isometry from M into M, a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a metric space. An isometric surjective linear operator on a Hilbert space is called a unitary operator, let X and Y be metric spaces with metrics dX and dY. A map ƒ, X → Y is called an isometry or distance preserving if for any a, b ∈ X one has d Y = d X. An isometry is automatically injective, otherwise two points, a and b, could be mapped to the same point, thereby contradicting the coincidence axiom of the metric d. This proof is similar to the proof that an order embedding between partially ordered sets is injective, clearly, every isometry between metric spaces is a topological embedding. A global isometry, isometric isomorphism or congruence mapping is a bijective isometry, like any other bijection, a global isometry has a function inverse. The inverse of an isometry is also a global isometry. Two metric spaces X and Y are called if there is a bijective isometry from X to Y. The set of bijective isometries from a space to itself forms a group with respect to function composition. This term is often abridged to simply isometry, so one should care to determine from context which type is intended. Any reflection, translation and rotation is a global isometry on Euclidean spaces, the map x ↦ | x | in R is a path isometry but not an isometry. Note that unlike an isometry, it is not injective, the isometric linear maps from Cn to itself are given by the unitary matrices. Given two normed vector spaces V and W, an isometry is a linear map f, V → W that preserves the norms. Linear isometries are distance-preserving maps in the above sense and they are global isometries if and only if they are surjective. By the Mazur-Ulam theorem, any isometry of normed spaces over R is affine. Note that ε-isometries are not assumed to be continuous, the restricted isometry property characterizes nearly isometric matrices for sparse vectors
27.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
28.
Star network
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Star networks are one of the most common computer network topologies. In its simplest form, a star network consists of one node, typically a switch or hub. In star topology, every node is connected to a central node, the switch is the server and the peripherals are the clients. A star network is an implementation of a spoke–hub distribution paradigm in computer networks, thus, the hub and leaf nodes, and the transmission lines between them, form a graph with the topology of a star. Data on a star passes through the hub, switch. The hub, switch, or concentrator manages and controls all functions of the network and it also acts as a repeater for the data flow. This configuration is common with twisted pair cable and optical fibre cable, however, it can also be used with coaxial cable. The star topology reduces the impact of a failure by connecting all of the systems to a central node. All peripheral nodes may communicate with all others by transmitting to, and receiving from. Advantages If one node or its connection breaks it doesn’t affect the other computers and their connections
29.
Computer network
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A computer network or data network is a telecommunications network which allows nodes to share resources. In computer networks, networked computing devices exchange data with other using a data link. The connections between nodes are established using either cable media or wireless media, the best-known computer network is the Internet. Network computer devices that originate, route and terminate the data are called network nodes, nodes can include hosts such as personal computers, phones, servers as well as networking hardware. Two such devices can be said to be networked together when one device is able to exchange information with the other device, Computer networks differ in the transmission medium used to carry their signals, communications protocols to organize network traffic, the networks size, topology and organizational intent. In most cases, application-specific communications protocols are layered over other more general communications protocols and this formidable collection of information technology requires skilled network management to keep it all running reliably. The chronology of significant computer-network developments includes, In the late 1950s, in 1960, the commercial airline reservation system semi-automatic business research environment went online with two connected mainframes. Licklider developed a group he called the Intergalactic Computer Network. In 1964, researchers at Dartmouth College developed the Dartmouth Time Sharing System for distributed users of computer systems. The same year, at Massachusetts Institute of Technology, a group supported by General Electric and Bell Labs used a computer to route. Throughout the 1960s, Leonard Kleinrock, Paul Baran, and Donald Davies independently developed network systems that used packets to transfer information between computers over a network, in 1965, Thomas Marill and Lawrence G. Roberts created the first wide area network. This was an precursor to the ARPANET, of which Roberts became program manager. Also in 1965, Western Electric introduced the first widely used telephone switch that implemented true computer control, in 1972, commercial services using X.25 were deployed, and later used as an underlying infrastructure for expanding TCP/IP networks. In July 1976, Robert Metcalfe and David Boggs published their paper Ethernet, Distributed Packet Switching for Local Computer Networks, in 1979, Robert Metcalfe pursued making Ethernet an open standard. In 1976, John Murphy of Datapoint Corporation created ARCNET, a network first used to share storage devices. In 1995, the transmission speed capacity for Ethernet increased from 10 Mbit/s to 100 Mbit/s, by 1998, Ethernet supported transmission speeds of a Gigabit. Subsequently, higher speeds of up to 100 Gbit/s were added, the ability of Ethernet to scale easily is a contributing factor to its continued use. Providing access to information on shared storage devices is an important feature of many networks, a network allows sharing of files, data, and other types of information giving authorized users the ability to access information stored on other computers on the network
30.
Distributed computing
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Distributed computing is a field of computer science that studies distributed systems. A distributed system is a model in which components located on networked computers communicate and coordinate their actions by passing messages, the components interact with each other in order to achieve a common goal. Three significant characteristics of distributed systems are, concurrency of components, lack of a global clock, examples of distributed systems vary from SOA-based systems to massively multiplayer online games to peer-to-peer applications. A computer program that runs in a system is called a distributed program. There are many alternatives for the message passing mechanism, including pure HTTP, RPC-like connectors, Distributed computing also refers to the use of distributed systems to solve computational problems. In distributed computing, a problem is divided into many tasks, each of which is solved by one or more computers, which communicate with each other by message passing. The terms are used in a much wider sense, even referring to autonomous processes that run on the same physical computer. The entities communicate with each other by message passing, a distributed system may have a common goal, such as solving a large computational problem, the user then perceives the collection of autonomous processors as a unit. Other typical properties of distributed systems include the following, The system has to tolerate failures in individual computers. The structure of the system is not known in advance, the system may consist of different kinds of computers and network links, each computer has only a limited, incomplete view of the system. Each computer may know one part of the input. Distributed systems are groups of networked computers, which have the goal for their work. The terms concurrent computing, parallel computing, and distributed computing have a lot of overlap, the same system may be characterized both as parallel and distributed, the processors in a typical distributed system run concurrently in parallel. Parallel computing may be seen as a tightly coupled form of distributed computing. In distributed computing, each processor has its own private memory, Information is exchanged by passing messages between the processors. The figure on the right illustrates the difference between distributed and parallel systems, figure shows a parallel system in which each processor has a direct access to a shared memory. The situation is complicated by the traditional uses of the terms parallel and distributed algorithm that do not quite match the above definitions of parallel. The use of concurrent processes that communicate by message-passing has its roots in operating system architectures studied in the 1960s, the first widespread distributed systems were local-area networks such as Ethernet, which was invented in the 1970s
31.
Tropical geometry
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Tropical geometry is a relatively new area in mathematics, which might loosely be described as a piece-wise linear or skeletonized version of algebraic geometry. This has been motivated by the applications to algebraic geometry found by Grigory Mikhalkin. The adjective tropical in the name of the area was coined by French mathematicians in honor of the Hungarian-born Brazilian mathematician Imre Simon, jean-Eric Pin attributes the coinage to Dominique Perrin, whereas Simon himself attributes the word to Christian Choffrut. We will use the min convention, that addition is classical minimum. It is also possible to cast the whole subject in terms of the max convention, negating throughout, the basic ideas of tropical analysis have been developed independently in the same notations by mathematicians working in various fields. In 1987 V. P. Maslov introduced a version of integration procedure. He also noticed that the Legendre transformation and solutions of the Hamilton-Jacobi equation are linear operations in the tropical sense, the tropical semiring is a semiring, with the operations as follows, x ⊕ y = min, x ⊗ y = x + y. Tropical exponentiation is defined in the way as iterated tropical products. A monomial of variables in this semiring is a linear map, a polynomial in the semiring is the minimum of a finite number of such monomials, and is therefore a concave, continuous, piecewise linear function. The set of points where a tropical polynomial F is non-differentiable is called its associated tropical hypersurface, there are two important characterizations of these objects, Tropical hypersurfaces are exactly the rational polyhedral complexes satisfying a zero-tension condition. Tropical surfaces are exactly the non-Archimedean amoebas over an algebraically closed non-Archimedean field K and these two characterizations provide a dictionary between combinatorics and algebra. Such a dictionary can be used to take an algebraic problem, the tropical hypersurface can be generalized to a tropical variety by taking the non-Archimedean amoeba of ideals I in K instead of polynomials. The tropical variety of an ideal I equals the intersection of the tropical hypersurfaces associated to every polynomial in I and this intersection can be chosen to be finite. There are a number of articles and surveys on tropical geometry, the study of tropical curves is particularly well developed. Tropical geometry was used by economist Paul Klemperer to design auctions used by the Bank of England during the crisis in 2007. Shiozawa defined subtropical algebra as max-times or min-times semiring and he found that Ricardian trade theory can be interpreted as subtropical convex algebra. A tropical counterpart of Abel-Jacobi map can be applied to a crystal design, the weights in a weighted finite-state transducer are often required to be a tropical semiring. Max-plus algebra Amini, Omid, Baker, Matthew, Faber, Xander, Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13,2011
32.
Maria Chudnovsky
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Maria Chudnovsky is an Israeli-American mathematician working on graph theory and combinatorial optimization. She is a 2012 MacArthur Fellow, Chudnovsky is a professor in the department of mathematics at Princeton University. She grew up in Russia and Israel, studying at the Technion, after postdoctoral research at the Clay Mathematics Institute, she became an assistant professor at Princeton University in 2005, and moved to Columbia University in 2006. By 2014, she was the Liu Family Professor of Industrial Engineering and she returned to Princeton as a professor of mathematics in 2015. She is a citizen of Israel and a permanent resident of the USA, in 2012, she married Daniel Panner, a viola player who teaches at Mannes College The New School for Music and the Juilliard School. They have a son named Rafael, other research contributions of Chudnovsky include co-authorship of the first polynomial time algorithm for recognizing perfect graphs, and of a structural characterization of the claw-free graphs. Chudnovsky, Maria, Cornuéjols, Gérard, Liu, Xinming, Seymour, Paul, Vušković, Kristina, Recognizing Berge graphs, Combinatorica,25, 143–186, doi,10. 1007/s00493-005-0012-8, MR2127609. Chudnovsky, Maria, Seymour, Paul, The structure of graphs, Surveys in Combinatorics 2005, London Mathematical Society Lecture Note Series,327. Press, pp. 153–171, doi,10. 1017/CBO9780511734885.008, MR2187738. Chudnovsky, Maria, Robertson, Neil, Seymour, Paul, Thomas, Robin, The strong perfect graph theorem, Annals of Mathematics,164, 51–229, in 2004 Chudnovsky was named one of the “Brilliant 10” by Popular Science magazine. Her work on the perfect graph theorem won for her and her co-authors the 2009 Fulkerson Prize. In 2012 she was awarded an award under the MacArthur Fellows Program. Chudnovskys home page at Princeton University LinkedIn account
33.
Paul Seymour (mathematician)
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Paul Seymour is currently a professor at Princeton University, half in the department of mathematics and half in the program in applied and computational math. His research interest is in mathematics, especially graph theory. Many of his recent papers are available from his website and he won a Sloan Fellowship in 1983, and the Ostrowski Prize in 2004, and won the Fulkerson Prize in 1979,1994,2006 and 2009, and the Pólya Prize in 1983 and 2004. He received a doctorate from the University of Waterloo in 2008. Seymour was born in Plymouth, Devon, England and he was a day student at Plymouth College, and then studied at Exeter College, Oxford, gaining a BA degree in 1971, and D. Phil in 1975. From 1983 until 1996, he was at Bellcore, Morristown and he was also an adjunct professor at Rutgers University from 1984–1987 and at the University of Waterloo from 1988–1993. He became professor at Princeton University in 1996 and he is Editor-in-Chief for the Journal of Graph Theory. He married Shelley MacDonald of Ottawa in 1979, and they have two children, Amy and Emily, the couple separated amicably in 2007. His brother Leonard W. Seymour is Professor of gene therapy at Oxford University, combinatorics in Oxford in the 1970s was dominated by matroid theory, due to the influence of Dominic Welsh and Aubrey William Ingleton. Much of Seymours early work, up to about 1980, was on matroid theory, in 1980 he moved to Ohio State University, and began work with Neil Robertson. In about 1990 Robin Thomas began to work with Robertson and Seymour, Seymours student Maria Chudnovsky joined them in 2001, and in 2002 the four jointly proved the conjecture. Robertson–Seymour theorem Paul Seymour home page at Princeton University Paul Seymour at the Mathematics Genealogy Project
34.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain content is freely available to anyone. William G. Bowen, president of Princeton University from 1972 to 1988, JSTOR originally was conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term, online access and full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution, JSTOR was initiated in 1995 at seven different library sites, and originally encompassed ten economics and history journals. JSTOR access improved based on feedback from its sites. Special software was put in place to make pictures and graphs clear, with the success of this limited project, Bowen and Kevin Guthrie, then-president of JSTOR, wanted to expand the number of participating journals. They met with representatives of the Royal Society of London and an agreement was made to digitize the Philosophical Transactions of the Royal Society dating from its beginning in 1665, the work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially, until January 2009 JSTOR operated as an independent, self-sustaining nonprofit organization with offices in New York City and in Ann Arbor, Michigan. JSTOR content is provided by more than 900 publishers, the database contains more than 1,900 journal titles, in more than 50 disciplines. Each object is identified by an integer value, starting at 1. In addition to the site, the JSTOR labs group operates an open service that allows access to the contents of the archives for the purposes of corpus analysis at its Data for Research service. This site offers a facility with graphical indication of the article coverage. Users may create focused sets of articles and then request a dataset containing word and n-gram frequencies and they are notified when the dataset is ready and may download it in either XML or CSV formats. The service does not offer full-text, although academics may request that from JSTOR, JSTOR Plant Science is available in addition to the main site. The materials on JSTOR Plant Science are contributed through the Global Plants Initiative and are only to JSTOR
35.
Bruce Reed (mathematician)
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Bruce Alan Reed FRSC is a Canadian mathematician and computer scientist, the Canada Research Chair in Graph Theory and a professor of computer science at McGill University. His research is primarily in graph theory, Reed earned his Ph. D. in 1986 from McGill, under the supervision of Vašek Chvátal. Before returning to McGill as a Canada Research Chair, Reed held positions at the University of Waterloo, Carnegie Mellon University, Reed was elected as a fellow of the Royal Society of Canada in 2009, and is the recipient of the 2013 CRM-Fields-PIMS Prize. Reeds thesis research concerned perfect graphs, with Michael Molloy, he is the author of a book on graph coloring and the probabilistic method. He was a speaker at the International Congress of Mathematicians in 2002. Home page at McGill Bruce A. Reed at DBLP Bibliography Server Bruce Reed publications indexed by Google Scholar
36.
ArXiv
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In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14,1991, arXiv. org passed the half-million article milestone on October 3,2008, by 2014 the submission rate had grown to more than 8,000 per month. The arXiv was made possible by the low-bandwidth TeX file format, around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Additional modes of access were added, FTP in 1991, Gopher in 1992. The term e-print was quickly adopted to describe the articles and its original domain name was xxx. lanl. gov. Due to LANLs lack of interest in the rapidly expanding technology, in 1999 Ginsparg changed institutions to Cornell University and it is now hosted principally by Cornell, with 8 mirrors around the world. Its existence was one of the factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists regularly upload their papers to arXiv. org for worldwide access, Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv. The annual budget for arXiv is approximately $826,000 for 2013 to 2017, funded jointly by Cornell University Library, annual donations were envisaged to vary in size between $2,300 to $4,000, based on each institution’s usage. As of 14 January 2014,174 institutions have pledged support for the period 2013–2017 on this basis, in September 2011, Cornell University Library took overall administrative and financial responsibility for arXivs operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it was supposed to be a three-hour tour, however, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. The lists of moderators for many sections of the arXiv are publicly available, additionally, an endorsement system was introduced in 2004 as part of an effort to ensure content that is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, new authors from recognized academic institutions generally receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for allegedly restricting scientific inquiry, perelman appears content to forgo the traditional peer-reviewed journal process, stating, If anybody is interested in my way of solving the problem, its all there – let them go and read about it. The arXiv generally re-classifies these works, e. g. in General mathematics, papers can be submitted in any of several formats, including LaTeX, and PDF printed from a word processor other than TeX or LaTeX. The submission is rejected by the software if generating the final PDF file fails, if any image file is too large. ArXiv now allows one to store and modify an incomplete submission, the time stamp on the article is set when the submission is finalized