On the Unit Graph of a Noncommutative Ring

Let $R$ be a ring (not necessary commutative) with non-zero identity. The unit graph of $R$, denoted by $G(R)$, is a graph with elements of $R$ as its vertices and two distinct vertices $a$ and $b$ are adjacent if and only if $a+b$ is a unit element of $R$. It was proved that if $R$ is a commutative ring and $\text{fm}$ is a maximal ideal of $R$ such that $\mathrm{\mid }R\mathrm{/}\text{fm}\mathrm{\mid }=2$, then $G(R)$ is a complete bipartite graph if and only if $(R,\text{fm})$ is a local ring. In this paper we generalize this result by showing that if $R$ is a ring (not necessary commutative), then $G(R)$ is a complete $r$-partite graph if and only if $(R, \fm)$isalocalringand$r=|R/m|=2^n$,forsome$n \in \N$or$R$ is a finite field. Among other results we show that if $R$ is a left Artinian ring, $2 \in U®$andthecliquenumberof$G®$isfinite,then$R$ is a finite ring.