Number of Edges of Regular Graph

Theorem

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

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.

$\blacksquare$

Proof of Corollary

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

$\blacksquare$