simple gragh G and complementary Graph

[asimov444]

If the simple graph G has v vertices and e edges, how many edges does G complement have?

The answer says, v(v-1)/2-e. I'm not quite sure how they derived with that answer. The definition says the complement G will have same vertices as G but not the edges.

 

[pka]

A complete graph on v vertices has \(\displaystyle \L\left( {\begin{array}{c}
v \\
2 \\
\end{array}} \right) = \frac{{v!}}{{\left( {2!} \right)\left( {v - 2} \right)!}} = \frac{{v\left( {v - 1} \right)}}{2}\)].
The complement of a graph has a edge if an only if that edge is absent in the graph itself.
Thus the number of edges in the complement is: \(\displaystyle \L\[
\frac{{v\left( {v - 1} \right)}}{2} - e
\]
.\)