The Diameter of Unit Graphs of Rings

Abstract

Let $R$ be a ring. The unit graph of $R$, denoted by $G(R)$, is the simple graph defined on all elements of $R$, and where two distinct vertices $x$ and $y$ are linked by an edge if and only if $x+y$ is a unit of $R$. The diameter of a simple graph $G$, denoted by $\operatorname{diam}(G)$, is the longest distance between all pairs of vertices of the graph $G$. In the present paper, we prove that for each integer $n \geq 1$, there exists a ring $R$ such that $n \leq \operatorname{diam}(G(R)) \leq 2n$. We also show that $\operatorname{diam}(G(R)) \in \{ 1,2,3,\infty \}$ for a ring $R$ with $R/J(R)$ self-injective and classify all those rings with $\operatorname{diam}(G(R)) = 1,2,3$ and $\infty$, respectively. This extends [12, Theorem 2 and Corollary 1].