A graph-theoretic approach for comparing dimensions of components in simply-graded algebras

Any simple group-grading of a finite dimensional complex algebra induces a natural family of digraphs. We prove that $|E\circ E^{\text{op}}\cup E^{\text{op}}\circ E|\geq |E|$ for any digraph $\Gamma =(V,E)$ without parallel edges, and deduce that for any simple group-grading, the dimension of the trivial component is maximal