Note that loops have two degrees, since one same edge incides twice on the same vertex.
We denote the minimum vertex degree in a graph with or , and the maximum with or .
Even Vertices, Odd Degrees
(corollary, from handshaking lemma)
In a graph, when the sum of the even degrees is even, then it must hold that the amount of vertices with odd degrees is even.
In a graph with no loops, the number of vertices with an odd degree is even.