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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 ]
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 graph with a loop having vertices labeled by degree. In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. [1]
In set theory and graph theory, denotes the set of n-tuples of elements of , that is, ordered sequences of elements that are not necessarily distinct. In the edge ( x , y ) {\displaystyle (x,y)} directed from x {\displaystyle x} to y {\displaystyle y} , the vertices x {\displaystyle x} and y {\displaystyle y} are called the endpoints of the ...
This is known as Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree.
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).
Here is a compilation put together in February that exemplifies Trump's "pull" and forceful style of handshake:
Another theorem that is commonly proven with a double counting argument states that every undirected graph contains an even number of vertices of odd degree. That is, the number of vertices that have an odd number of incident edges must be even.