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In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. For example, if there is a party of people who shake hands, the number of people who shake an odd number of other people's hands is even. [ 1 ]
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 of people, the number of people who have shaken hands with an odd ...
In more colloquial terms, in a party of people some of whom shake hands, an even number of people must have shaken an odd number of other people's hands; for this reason, the result is known as the handshaking lemma. To prove this by double counting, let () be the degree of vertex . The number of vertex-edge incidences in the graph may be ...
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 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).
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. Route inspection problem, search for the shortest path that visits all edges, possibly repeating edges if an Eulerian path does not exist.
The way the president shakes hands has been in the news as of late, especially those with French President Emmanuel Macron. Macron said his first shake with Trump was a "moment of truth." Another ...
It is modeled by an infinite ray, but violates Euler's handshaking lemma for finite graphs. However, it follows from the negative solution to the Entscheidungsproblem (by Alonzo Church and Alan Turing in the 1930s) that satisfiability of first-order sentences for graphs that are not constrained to be finite remains undecidable. It is also ...