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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 graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex. In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph ...
The degree or valency of a vertex is the number of edges that are incident to it, where a loop is counted twice. The degree of a graph is the maximum of the degrees of its vertices. In an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n(n − 1) / 2 .
In an empty graph, each vertex forms a component with one vertex and zero edges. [3] More generally, a component of this type is formed for every isolated vertex in any graph. [4] In a connected graph, there is exactly one component: the whole graph. [4] In a forest, every component is a tree. [5] In a cluster graph, every component is a ...
A one-vertex cut is called an articulation point or cut vertex. vertex set The set of vertices of a given graph G, sometimes denoted by V(G). vertices See vertex. Vizing 1. Vadim G. Vizing 2. Vizing's theorem that the chromatic index is at most one more than the maximum degree. 3.
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G).
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions).
In an undirected graph, the edge to the parent of a node should not be counted as a back edge, but finding any other already visited vertex will indicate a back edge. In the case of undirected graphs, only O(n) time is required to find a cycle in an n-vertex graph, since at most n − 1 edges can be tree edges.