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Math Help - Complements and Isometries

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    Complements and Isometries

    Show that if a simple graph G is isomorphic to its complement \overline{G}, then G has either 4k or 4k+1 vertices for some natural number k.
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    Quote Originally Posted by meggnog View Post
    Show that if a simple graph G is isomorphic to its complement \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 |V|=4k,~4k+1,~4k+2,\text{ or }4k+3. WHY?
    In a self complementary graph the number of edges in G must be same as the number of edges in \overline{G}. WHY?

    For which of the possible |V| is \dbinom{|V|}{2} even?
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