on the refinement of the unit and unitary cayley graphs of rings

Let r be a ring (not necessarily commutative) with nonzero identity. we define γ(r) to be the graph with vertex set r in which two distinct vertices x and y are adjacent if and only if there exist unit elements u,v of r such that x+uyv is a unit of r. in this paper, basic properties of γ(r) are studied. we investigate connectivity and the girth of γ(r), where r is a left artinian ring. we also determine when the graph γ(r) is a cycle graph. we prove that if γ(r)≅γ(mn(f)) then r≅mn(f), where r is a ring and f is a finite field. we show that if r is a finite commutative semisimple ring and s is a commutative ring such that γ(r)≅γ(s), then r≅s. finally, we obtain the spectrum of γ(r), where r is a finite commutative ring.