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

Authors : Huadong Su, Chunhong Huang

Pages : 1075-1084

Doi:10.15672/hujms.1256413

View : 111 | Download : 216

Publication Date : 2024-08-27

Article Type : Research Paper

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.
Keywords : finite commutative ring, comaximal graph, line graph, genus, induced subgraph