Number of Edges of Regular Graph
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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.
$\blacksquare$