Complement of Complete Bipartite Graph

Theorem
Let $$K_{p, q}$$ be a complete bipartite graph.

The complement of $$K_{p, q}$$ consists of a disconnected graph with two components:


 * The complete graph $$K_p$$;


 * The complete graph $$K_q$$.

Proof
By definition, the complete bipartite graph $$K_{p, q}$$ consists of two sets of vertices: $$A$$ of cardinality $$p$$, and $$B$$ of cardinality $$q$$, such that:


 * Every vertex in $$A$$ is adjacent to every vertex in $$B$$;


 * No vertex in $$A$$ is adjacent to any other vertex in $$A$$;


 * No vertex in $$B$$ is adjacent to any other vertex in $$B$$.

The complement of $$K_{p, q}$$ therefore must be a graph $$G$$ such that:


 * No vertex in $$A$$ is adjacent to any vertex in $$B$$;


 * Every vertex in $$A$$ is adjacent to every other vertex in $$A$$;


 * Every vertex in $$B$$ is adjacent to every other vertex in $$B$$.

The second and third of these conditions describes the complete graphs $$K_p$$ and $$K_q$$.

From the first of these conditions, it follows that $$G$$ comes in two disconnected graph components.

Hence the result.