Show that if a simple graph G is isomorphic to its complement $\displaystyle \overline{G}$, then G has either 4k or 4k+1 vertices for some natural number k.
In any simple graph the number of vertices will be $\displaystyle |V|=4k,~4k+1,~4k+2,\text{ or }4k+3$. WHY?
In a self complementary graph the number of edges in $\displaystyle G$ must be same as the number of edges in $\displaystyle \overline{G}$. WHY?
For which of the possible $\displaystyle |V|$ is $\displaystyle \dbinom{|V|}{2}$ even?