Number of Edges of Regular Graph

Theorem
An $r$-regular graph of order $$n$$ is of size $$\frac {n r} 2$$.

Corollary
There are no $r$-regular graph of order $$n$$ where both $$n$$ and $$r$$ are odd.

Proof
The size of a $r$-regular graph is its number of edges.

The order of a $r$-regular graph is its number of vertices.

The degree of each vertex of an $r$-regular graph is $$r$$.

Hence the total of all the degrees of an $r$-regular graph of order $$n$$ is $$nr$$.

The result follows directly from the Handshake Lemma.

Proof of Corollary
If $$n$$ and $$r$$ are both odd, then $$nr$$ is also odd, and hence $$\frac {nr} 2$$ is not integral.