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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 ...
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 vertex can reach a vertex (and is reachable from ) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with and ends with . In an undirected graph, reachability between all pairs of vertices can be determined by identifying the connected components of the graph.
In mathematics, calculus on finite weighted graphs is a discrete calculus for functions whose domain is the vertex set of a graph with a finite number of vertices and weights associated to the edges.
A vertex with a large degree also called a heavy node, results in a large diagonal entry in the Laplacian matrix dominating the matrix properties. Normalization is aimed to make the influence of such vertices more equal to that of other vertices, by dividing the entries of the Laplacian matrix by the vertex degrees.
A vertex with degree 1 is called a leaf vertex or end vertex or a pendant vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of trees in graph theory and especially trees as data structures .
The graph of the 3-3 duoprism (the line graph of ,) is perfect.Here it is colored with three colors, with one of its 3-vertex maximum cliques highlighted. In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph.
The Mycielski graph μ(G) contains G itself as a subgraph, together with n+1 additional vertices: a vertex u i corresponding to each vertex v i of G, and an extra vertex w. Each vertex u i is connected by an edge to w , so that these vertices form a subgraph in the form of a star K 1, n .