In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal.
In discrete 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 are represented by abstractions called vertices and each of the related pairs of vertices is called an edge. Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges.
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct. A directed path in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid.
In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number of disjoint paths that can be found between any pair of vertices. Proved by Karl Menger in 1927, it characterizes the connectivity of a graph. It is generalized by the max-flow min-cut theorem, which is a weighted, edge version, and which in turn is a special case of the strong duality theorem for linear programs.
In graph theory, a cograph, or complement-reducible graph, or P4-free graph, is a graph that can be generated from the single-vertex graph K1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K1 and is closed under complementation and disjoint union.
In graph theory, the Cartesian productG □ H of graphs G and H is a graph such that:
In graph theory, an edge contraction is an operation that removes an edge from a graph while simultaneously merging the two vertices that it previously joined. Edge contraction is a fundamental operation in the theory of graph minors. Vertex identification is a less restrictive form of this operation.
In computer graphics, a triangle mesh is a type of polygon mesh. It comprises a set of triangles that are connected by their common edges or vertices.
In graph theory, the hypercube graphQn is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Qn has 2n vertices, 2n – 1n edges, and is a regular graph with n edges touching each vertex.
In complexity theory and graph theory, induced subgraph isomorphism is an NP-complete decision problem that involves finding a given graph as an induced subgraph of a larger graph.
In graph theory, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs G1 and G2 and produces a graph H with the following properties:
In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as graphs in which some edges are labeled by matrices. The simplest trace diagrams represent the trace and determinant of a matrix. Several results in linear algebra, such as Cramer's Rule and the Cayley–Hamilton theorem, have simple diagrammatic proofs. They are closely related to Penrose's graphical notation.
In the mathematical discipline of graph theory, a graph C is a covering graph of another graph G if there is a covering map from the vertex set of C to the vertex set of G. A covering map f is a surjection and a local isomorphism: the neighbourhood of a vertex v in C is mapped bijectively onto the neighbourhood of in G.
In graph theory, the zig-zag product of regular graphs , denoted by , is a binary operation which takes a large graph and a small graph and produces a graph that approximately inherits the size of the large one but the degree of the small one. An important property of the zig-zag product is that if is a good expander, then the expansion of the resulting graph is only slightly worse than the expansion of .
In graph theory and computer science, the graph sandwich problem is a problem of finding a graph that belongs to a particular family of graphs and is "sandwiched" between two other graphs, one of which must be a subgraph and the other of which must be a supergraph of the desired graph.
In information theory, the graph entropy is a measure of the information rate achievable by communicating symbols over a channel in which certain pairs of values may be confused. This measure, first introduced by Körner in the 1970s, has since also proven itself useful in other settings, including combinatorics.
In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph.
In graph theory, a balanced hypergraph is a hypergraph that has several properties analogous to that of a bipartite graph.
A sparsity matroid is a mathematical structure that captures how densely a multigraph is populated with edges. To unpack this a little, sparsity is a measure of density of a graph that bounds the number of edges in any subgraph. The property of having a particular matroid as its density measure is invariant under graph isomorphisms and so it is a graph invariant.
