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Complement (graph Theory)
In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there.. The complement is not the set complement of the graph; only the edges are complemented. Definition Let be a simple graph and let consist of all 2-element subsets of . Then is the complement of , where is the relative complement of in . For directed graphs, the complement can be defined in the same way, as a directed graph on the same vertex set, using the set of all 2-element ordered pairs of in place of the set in the formula above. In terms of the adjacency matrix ''A'' of the graph, if ''Q'' is the adjacency matrix of the complete graph of the same number of vertices (i.e. all entries are unity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersen Graph Complement
Petersen is a common Danish patronymic surname, meaning ''"son of Peter"''. There are other spellings. Petersen may refer to: People In arts and entertainment * Adolf Dahm-Petersen, Norwegian voice specialist * Anja Petersen, German operatic soprano and university lecturer * Anker Eli Petersen, Faroese writer and artist * Ann Petersen, Belgian actress * Chris Petersen (born 1963), American child actor * Devon Petersen (born 1986), South African darts player * Elmer Petersen, American artist * Gustaf Munch-Petersen, Danish writer and painter * Joel Petersen, bass guitarist * John Hahn-Petersen, Danish actor * Josef Petersen, Danish novelist * Patrick Petersen, American actor * Paul Petersen, American movie actor, singer, novelist, and activist * Robert E. Petersen, publisher, auto museum founder * Robert Storm Petersen, Danish cartoonist, writer, animator, illustrator, painter and humorist * Sandy Petersen, American game designer * Uwe Fahrenkrog-Petersen, German musician * Wil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edgeless Graph
In the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph"). Order-zero graph The order-zero graph, , is the unique graph having no vertices (hence its order is zero). It follows that also has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including as a valid graph is useful depends on context. On the positive side, follows naturally from the usual set-theoretic definitions of a graph (it is the ordered pair for which the vertex and edge sets, and , are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined data structures is useful for defining the base case for recursion (by treating the null tree as the child of missi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
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 graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Graph
In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph (clique number). Equivalently stated in symbolic terms an arbitrary graph G=(V,E) is perfect if and only if for all S\subseteq V we have \chi(G =\omega(G . The perfect graphs include many important families of graphs and serve to unify results relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time. In addition, several important min-max theorems in combinatorics, such as Dilworth's theorem, can be expressed in terms of the perfection of certain associated graphs. A graph G is 1-perfect if and only if \chi(G)=\omega(G). Then, G is perfect if and only if every induced subgraph of G is 1-perfect. Properties * By the perfect graph theorem, a graph G is perfect if and on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called . The number of vertices in equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it. Terminology There are many synonyms for "cycle graph". These include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or ''n''-gon are also often used. The term ''n''-cycle is sometimes used in other settings. A cycle with an even number of vertices is called an even cycle; a cycle with an odd number of vertices is called an odd cycle. Properties A cycle graph is: * 2-edge colorable, if and only if it has an even number of vertices * 2-regular * 2-ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Graph
In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order such that the edges are where . Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph. A path is a particularly simple example of a tree, and in fact the paths are exactly the trees in which no vertex has degree 3 or more. A disjoint union of paths is called a linear forest. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See, for example, Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). As Dynkin diagrams In algebra, path graphs appear as the Dynkin diagrams of type A. As such, they classify the root system of type A and the Weyl group of ty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H'' : f \colon V(G) \to V(H) such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) are adjacent in ''H''. 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\simeq H. In the case when the bijection is a mapping of a graph onto itself, i.e., when ''G'' and ''H'' are one and the same graph, the bijection is called an automorphism of ''G''. If a graph is finite, we can prove it to be bijective by showing it is one-one/onto; no need to show both. Graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. A set of graphs isomorphic to each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-complementary Graph
In the mathematical field of graph theory, a self-complementary graph is a graph which is isomorphic to its complement. The simplest non-trivial self-complementary graphs are the path graph and the cycle graph. There is no known characterization of self-complementary graphs. Examples Every Paley graph is self-complementary. For example, the 3 × 3 rook's graph (the Paley graph of order nine) is self-complementary, by a symmetry that keeps the center vertex in place but exchanges the roles of the four side midpoints and four corners of the grid. All strongly regular self-complementary graphs with fewer than 37 vertices are Paley graphs; however, there are strongly regular graphs on 37, 41, and 49 vertices that are not Paley graphs. The Rado graph is an infinite self-complementary graph. Properties An self-complementary graph has exactly half number of edges of the complete graph, i.e., edges, and (if there is more than one vertex) it must have diameter either 2 or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claw-free Graph
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 ''K''1,3 (that is, a star graph comprising three edges, three leaves, and a central vertex). 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 claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph. Claw-free graphs were initially studied as a generalization of line graphs, and gained additional motivation through three key discoveries about them: the fact that all claw-free connected graphs of even order have perfect matchings, the discovery of polynomial time algorithms for finding maximum independent sets in claw-free graphs, and the characterization of claw-free perfect graphs., p. 88. They are the subject of hundreds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle-free Graph
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. By Turán's theorem, the ''n''-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible. Triangle finding problem The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with edges is triangle-free in time . Another approach is to find the trace of , where is the adjacency matrix of the graph. The trace is zero if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Automorphism
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 is a permutation of the vertex set , such that the pair of vertices form an edge if and only if the pair also form an edge. That is, it is a graph isomorphism from 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 Frucht's theorem, all groups can be represented as the automorphism group of a connected graph – indeed, of a cubic graph. Computational complexity Constructing the automorphism group is at least as difficult (in terms of its computational complexity) as solving the graph is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
