Number of Edges of Regular Graph

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

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.