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.