On the unitary one matching Bi-Cayley graph over finite rings

Abstract

Let [Formula: see text] be a finite ring (with nonzero identity) and let [Formula: see text] denote the unitary one-matching bi-Cayley graph over [Formula: see text] In this paper, we calculate the chromatic, edge chromatic, clique and independent numbers of [Formula: see text] and we show that the graph [Formula: see text] is a strongly regular graph if and only if [Formula: see text]. Also, we study the perfectness of [Formula: see text] Moreover, we prove that if [Formula: see text] and [Formula: see text] then [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are Jacobson radicals of [Formula: see text] and [Formula: see text], respectively. Furthermore, for a finite field [Formula: see text] with [Formula: see text] and a ring [Formula: see text], we prove that if [Formula: see text] where [Formula: see text] is an integer, then [Formula: see text] and so [Formula: see text] is a semisimple ring.