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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]
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]
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 ...
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.
Pages in category "Lemmas in graph theory" The following 5 pages are in this category, out of 5 total. ... Handshaking lemma; K. Kőnig's lemma; S. Szemerédi ...
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.
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.
The algorithm addresses the problem that T is not a tour by identifying all the odd degree vertices in T; since the sum of degrees in any graph is even (by the Handshaking lemma), there is an even number of such vertices. The algorithm finds a minimum-weight perfect matching M among the odd-degree ones.