of the graph (meaning it is a linear combination of powers of A).

Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix . If G is not bipartite, then

Existence

There exists a -regular graph of order if and only if the natural numbers and satisfy the inequality and that is even.

Proof: If a graph with vertices is -regular, then the degree of any vertex v cannot exceed the number of vertices different from v,