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A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
A directed graph is weakly connected (or just connected [9]) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph. A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices (x, y).
Graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph) {,} = {} which is not in {{,},}. To allow loops, the definitions must be expanded.
An undirected acyclic graph is the same thing as a forest. An acyclic directed graph, which is a digraph without directed cycles, is often called a directed acyclic graph, especially in computer science. [2] 2. An acyclic coloring of an undirected graph is a proper coloring in which every two color classes induce a forest. [3] adjacency matrix
A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. [2] A directed tree, [3] oriented tree, [4] [5] polytree, [6] or singly connected network [7] is a directed acyclic graph (DAG) whose underlying undirected graph is ...
An undirected graph. In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented. The unoriented incidence matrix (or simply incidence matrix ) of an undirected graph is a n × m {\displaystyle n\times m} matrix B , where n and m are the numbers of vertices and edges respectively, such that
A directed graph is called an oriented graph if none of its pairs of vertices is linked by two mutually symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of (x, y) and (y, x) may be arrows of the graph). [1] A tournament is an orientation of a complete graph.
An ancestral graph is a further extension, having directed, bidirected and undirected edges. [4] Random field techniques A Markov random field, also known as a Markov network, is a model over an undirected graph. A graphical model with many repeated subunits can be represented with plate notation.