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

Year 2024, Volume: 53 Issue: 4, 1075 - 1084, 27.08.2024

Huadong Su , Chunhong Huang

https://doi.org/10.15672/hujms.1256413

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

Thanks

This research was supported by the Natural Science Foundation of China (Grant No. 12261001) and the Guangxi Natural Science Foundation (Grant No. 2021GXNSFAA220043) and High-level talents for scientific research of Beibu Gulf University (2020KYQD07).

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