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Diamond Graph
In the mathematical field of graph theory, the diamond graph is a planar, undirected graph with 4 vertices and 5 edges. It consists of a complete graph minus one edge. The diamond graph has radius 1, diameter 2, girth 3, chromatic number 3 and chromatic index 3. It is also a 2- vertex-connected and a 2- edge-connected, graceful, Hamiltonian graph. Diamond-free graphs and forbidden minor A graph is diamond-free if it has no diamond as an induced subgraph. The triangle-free graphs are diamond-free graphs, since every diamond contains a triangle. The diamond-free graphs are locally clustered: that is, they are the graphs in which every neighborhood is a cluster graph. Alternatively, a graph is diamond-free if and only if every pair of maximal cliques in the graph shares at most one vertex. The family of graphs in which each connected component is a cactus graph is downwardly closed under graph minor operations. This graph family may be characteriz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Subgraph
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, from the original graph, connecting pairs of vertices in that subset. Definition Formally, let G=(V,E) be any graph, and let S\subseteq V be any subset of vertices of . Then the induced subgraph G is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S . That is, for any two vertices u,v\in S , u and v are adjacent in G if and only if they are adjacent in G . 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 , or (if context makes the choice of G unambiguous) the induced subgraph of S . Examples Important types of induced subgraphs include the following. * Induced paths are induced subgraphs that are paths. The shortest path between any two vertices in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the correspondi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, generated by a single element. That is, it is a set (mathematics), set of Inverse element, invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer Exponentiation, power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a ''Generating set of a group, generator'' of the group. Every infinite cyclic group is isomorphic to the additive group \Z, the integers. Every finite cyclic group of Order (group theory), order n is isomorphic to the additive group of Quotient group, Z/''n''Z, the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Product Of Groups
In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G \oplus H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups. Definition Given groups (with operation ) and (with operation ), the direct product is defined as follows: The resulting algebraic object satisfies the axioms for a group. Specifically: ;Associativity: The binary operation on is associative. ;Identity: The direct product has an identity element, namely , where is the identi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoforest
In graph theory, a pseudoforest is an undirected graphThe kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph. in which every Connected component (graph theory), connected component has at most one Cycle (graph theory), cycle. That is, it is a system of Vertex (graph theory), vertices and Edge (graph theory), edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest. The names are justified by analogy to the more commonly studied Tree (graph theory), trees and Forest (graph theory), forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan. attribute the study of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butterfly Graph
In the mathematics, mathematical field of graph theory, the butterfly graph (also called the bowtie graph and the hourglass graph) is a planar graph, planar, undirected graph with 5 Vertex (graph theory), vertices and 6 edges. It can be constructed by joining 2 copies of the cycle graph with a common vertex and is therefore Graph isomorphism, isomorphic to the friendship graph . The butterfly graph has graph diameter, diameter 2 and girth (graph theory), girth 3, radius 1, chromatic number 3, chromatic index 4 and is both Eulerian graph, Eulerian and a penny graph (this implies that it is unit distance graph, unit distance and planar graph, planar). It is also a 1-k-vertex-connected graph, vertex-connected graph and a 2-k-edge-connected graph, edge-connected graph. There are only three Graceful labeling, non-graceful simple graphs with five vertices. One of them is the butterfly graph. The two others are cycle graph and the complete graph . Bowtie-free grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forbidden Minor
Forbidden may refer to: Science * Forbidden mechanism, a spectral line associated with absorption or emission of photons Films * ''Forbidden'' (1919 film), directed by Phillips Smalley and Lois Weber * ''Forbidden'' (1932 film), directed by Frank Capra * ''Forbidden'' (1949 film), directed by George King * ''Forbidden'' (1953 film), directed by Rudolph Maté * ''Forbidden'' (''Proibito''), a 1954 Italian film directed by Mario Monicelli * ''Forbidden'' (1984 film), directed by Anthony Page * '' The Forbidden'', a 2018 Uganda film Literature * ''Forbidden'' (Cooney novel), a 1994 novel by Caroline B. Cooney * ''Forbidden'' (Dekker and Lee novel), 2011 novel by Ted Dekker and Tosca Lee * ''Forbidden'', a 2010 novel by Tabitha Suzuma * "The Forbidden", short story by Clive Barker, from the Books of Blood Music * Forbidden (band), an American thrash metal band * ''Forbidden'' (Black Sabbath album) (1995), also the title track * ''Forbidden'' (Todrick Hall album) (2018), al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Minor
In graph theory, an undirected graph is called a minor of the graph if can be formed from by deleting edges, vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph nor the complete bipartite graph ., p. 77; . The Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs that is preserved by deletions and edge contractions., theorem 4, p. 78; . For every fixed graph , it is possible to test whether is a minor of an input graph in polynomial time; together with the forbidden minor characterization this implies that every graph property preserved by deletions and contractions may be recognized in polynomial time. Other results and conjectures involving graph minors include the graph structure theorem, according to which the graphs that do not have as a minor may be formed by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cactus Graph
In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cacti) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle. Properties Cacti are outerplanar graphs. Every pseudotree is a cactus. A nontrivial graph is a cactus if and only if every block is either a simple cycle or a single edge. The family of graphs in which each component is a cactus is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor, the four-vertex diamond graph formed by removing an edge from the complete graph ''K''4. Triangular cactus A triangular cactus is a special type of cactus graph such that each cycle has length three and each edge belongs to a cycle. For instance, the friendship graphs, graphs formed f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Component (graph Theory)
In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components. The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices. In random graphs, a frequently occurring phenomenon is the incidence of a giant component, one component that is significantly larger than the others; and of a percolation threshold, an edge probability above which a giant component exists and below which it does not. The components of a graph can be constructed in linear time, and a special case of the problem, connected-component labeling, is a basic technique in image analysis. Dy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
