On the unitary Cayley graph of a ring

Abstract

Let R be a ring with identity. The unitary Cayley graph of a ring R, denoted by G R, is the graph, whose vertex set is R, and in which {x; y} is an edge if and only if x-y is a unit of R. In this paper we find chromatic, clique and independence number of G R, where R is a finite ring. Also, we prove that if G R≃G S, then G R/JR≃G S/JS, where JR and JS are Jacobson radicals of R and S, respectively. Moreover, we prove if G R≃G Mn(F) then R≃M n(F), where R is a ring and F is a finite field. Finally, let R and S be finite commutative rings, we show that if G R≃G S, then R=JR ' S=JS.