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 $Gamma(R)$ 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 $Gamma(R)$ are studied. We investigate connectivity and the girth of $Gamma(R)$, where $R$ is a left Artinian ring. We also determine when the graph $Gamma(R)$ is a cycle graph. We prove that if $Gamma(R)congGamma(M_{n}(F))$ then $Rcong M_{n}(F)$, 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 $Gamma(R)congGamma(S)$, then $Rcong S$. Finally, we obtain the spectrum of $Gamma(R)$, where $R$ is a finite commutative ring.

Keywords:

Rings , Matrix rings , Jacobson radical , Unit graphs , Unitary Cayley graphs