Subgraph of unitary Cayley graph of matrix algebras induced by idempotent matrices

Abstract

In this work, we are interested in the subgraph of unitary Cayley graph of a matrix algebra over a finite field induced by the set of idempotent matrices. We study its structure and completely obtain all connected components with the degree on each vertex. Most of them turn out to be a regular bipartite graph such that each partite set has the same number of vertices. Our main tools are combinatorial results on matrix algebras. Moreover, we compute the clique number, chromatic number and independence number of the graph.