Number of Edges of Regular Graph/Corollary

Corollary to Number of Edges of Regular Graph

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

Proof

From Number of Edges of Regular Graph, an $r$-regular graph of order $n$ is of size $\dfrac {n r} 2$.

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

$\blacksquare$