On the Unitary Cayley Graph of a Ring

Let R be a ring with identity. The unitary Cayley graph of a ring R, denoted by GR, 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 GR, where R is a finite ring. Also, we prove that if GR ' GS , then GR/JR ' GS/JS , where JR and JS are Jacobson radicals of R and S, respectively. Moreover, we prove if GR ' GMn(F ) then R ' Mn(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 GR ' GS , then R/JR ' S/JS.