%0 Journal Article %T ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS %J Journal of Algebraic Systems %I Shahrood University of Technology %Z 2345-5128 %A Rezagholibeigi, M. %A Naghipour, A. R. %D 2019 %\ 09/01/2019 %V 7 %N 1 %P 51-68 %! ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS %K Rings %K Matrix rings %K Jacobson radical %K Unit graphs %K Unitary Cayley graphs %R 10.22044/jas.2018.6939.1340 %X 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 $\Gamma(R)\cong\Gamma(M_{n}(F))$ then $R\cong M_{n}(F)$, 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 $\Gamma(R)\cong\Gamma(S)$, then $R\cong S$. Finally, we obtain the spectrum of $\Gamma(R)$, where $R$ is a finite commutative ring. %U https://jas.shahroodut.ac.ir/article_1439_ebb88b5d1aeb02a640e428d56a015c93.pdf