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For graphs that are allowed to contain loops connecting a vertex to itself, a loop should be counted as contributing two units to the degree of its endpoint for the purposes of the handshaking lemma. [2] Then, the handshaking lemma states that, in every finite graph, there must be an even number of vertices for which is an odd number. [1]
It is a class of search problems that can be shown to be total by an application of the handshaking lemma: any undirected graph that has a vertex whose degree is an odd number must have some other vertex whose degree is an odd number. This observation means that if we are given a graph and an odd-degree vertex, and we are asked to find some ...
The degree sum formula states that, given a graph = (,), = | |. The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, which is to prove that in any group ...
The total degree is the sum of the degrees of all vertices; by the handshaking lemma it is an even number. The degree sequence is the collection of degrees of all vertices, in sorted order from largest to smallest. In a directed graph, one may distinguish the in-degree (number of incoming edges) and out-degree (number of outgoing edges). [2] 2.
From the handshaking lemma, a k-regular graph with odd k has an even number of vertices. A theorem by Nash-Williams says that every k ‑regular graph on 2k + 1 vertices has a Hamiltonian cycle. Let A be the adjacency matrix of a graph. Then the graph is regular if and only if = (, …,) is an eigenvector of A. [2]
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 ) .
Eulerian matroid, an abstract generalization of Eulerian graphs; Five room puzzle; Handshaking lemma, proven by Euler in his original paper, showing that any undirected connected graph has an even number of odd-degree vertices; Hamiltonian path – a path that visits each vertex exactly once.
In mathematics, the Lindström–Gessel–Viennot lemma provides a way to count the tuples of non-intersecting lattice paths, or, more generally, paths on a directed graph. It was proved by Gessel–Viennot in 1985, based on previous work of Lindström published in 1973. The lemma is named after Bernt Lindström, Ira Gessel and Gérard Viennot.