Finite commutative rings whose line graphs of comaximal graphs have genus at most two

Abstract

Let $R$ be a ring with identity. The comaximal graph of $R$, denoted by $\Gamma(R)$, is a simple graph with vertex set $R$ and two different vertices $a$ and $b$ are adjacent if and only if $aR+bR=R$. Let $\Gamma_{2}(R)$ be a subgraph of $\Gamma(R)$ induced by $R\backslash\{U(R)\cup J(R)\}$. In this paper, we investigate the genus of the line graph $L(\Gamma(R))$ of $\Gamma(R)$ and the line graph $L(\Gamma_{2}(R))$ of $\Gamma_2(R)$. All finite commutative rings whose genus of $L(\Gamma(R))$ and $L(\Gamma_{2}(R))$ are 0, 1, 2 are completely characterized, respectively.