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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 sum of degrees of all six vertices is 2 + 3 + 2 + 3 + 3 + 1 = 14, twice the number of edges. 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.
A graph meeting the conditions of Ore's theorem, and a Hamiltonian cycle in it. There are two vertices with degree less than n/2 in the center of the drawing, so the conditions for Dirac's theorem are not met. However, these two vertices are adjacent, and all other pairs of vertices have total degree at least seven, the number of vertices.
where g is the degree sum, e is the number of edges in the given graph, v is the number of vertices, and c is the number of connected components. The degree sum of a hypergraph is the sum of the degrees of all the vertices, reducing to 2e for a simple graph, or ke for a k-uniform hypergraph. This formula is symmetric between vertices and edges ...
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).
The inequality between the sum of the largest degrees and the sum of the remaining degrees can be established by double counting: the left side gives the numbers of edge-vertex adjacencies among the highest-degree vertices, each such adjacency must either be on an edge with one or two high-degree endpoints, the () term on the right gives the ...
The degree sequence of a directed graph is the list of its indegree and outdegree pairs; for the above example we have degree sequence ((2, 0), (2, 2), (0, 2), (1, 1)). The degree sequence is a directed graph invariant so isomorphic directed graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely ...
The number of vertices must be doubled because each undirected edge corresponds to two directed arcs and thus the degree of a vertex in the directed graph is twice the degree in the undirected graph. Rahman– Kaykobad (2005) — A simple graph with n vertices has a Hamiltonian path if, for every non-adjacent vertex pairs the sum of their ...