Graph labelingIn the mathematical discipline of graph theory, a graph labelling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph. Formally, given a graph G = (V, E), a vertex labelling is a function of V to a set of labels; a graph with such a function defined is called a vertex-labeled graph. Likewise, an edge labelling is a function of E to a set of labels. In this case, the graph is called an edge-labeled graph. When the edge labels are members of an ordered set (e.
Connectivity (graph theory)In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v.
Algebraic graph theoryAlgebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants. The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra.
DerangementIn combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position. In other words, a derangement is a permutation that has no fixed points. The number of derangements of a set of size n is known as the subfactorial of n or the n-th derangement number or n-th de Montmort number (after Pierre Remond de Montmort). Notations for subfactorials in common use include !n, Dn, dn, or n¡.
Topological graph theoryIn mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies immersions of graphs. Embedding a graph in a surface means that we want to draw the graph on a surface, a sphere for example, without two edges intersecting. A basic embedding problem often presented as a mathematical puzzle is the three utilities problem.
Unimodular latticeIn geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1. The E8 lattice and the Leech lattice are two famous examples. A lattice is a free abelian group of finite rank with a symmetric bilinear form (·, ·). The lattice is integral if (·,·) takes integer values. The dimension of a lattice is the same as its rank (as a Z-module).
Induced subgraphIn the mathematical field of 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. Formally, let be any graph, and let be any subset of vertices of G. Then the induced subgraph is the graph whose vertex set is and whose edge set consists of all of the edges in that have both endpoints in . That is, for any two vertices , and are adjacent in if and only if they are adjacent in .
Hadwiger conjecture (graph theory)In graph theory, the Hadwiger conjecture states that if is loopless and has no minor then its chromatic number satisfies . It is known to be true for . The conjecture is a generalization of the four-color theorem and is considered to be one of the most important and challenging open problems in the field. In more detail, if all proper colorings of an undirected graph use or more colors, then one can find disjoint connected subgraphs of such that each subgraph is connected by an edge to each other subgraph.
Cycle (graph theory)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. A graph without cycles is called an acyclic graph. A directed graph without directed cycles is called a directed acyclic graph. A connected graph without cycles is called a tree. A circuit is a non-empty trail in which the first and last vertices are equal (closed trail).
Network scienceNetwork science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks, considering distinct elements or actors represented by nodes (or vertices) and the connections between the elements or actors as links (or edges). The field draws on theories and methods including graph theory from mathematics, statistical mechanics from physics, data mining and information visualization from computer science, inferential modeling from statistics, and social structure from sociology.
