On the graph $G_P(R)$ over commutative ring R

期刊论文
发表日期：

2024年5月30日

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B. Biswas,
S. Kar

摘要

Let R be a commutative ring with identity 1. Then the graph of R, denoted by $G_P(R)$ $G_{P} (R)$ which is defined as the vertices are the elements of R and any two distinct elements a and b are adjacent if and only if the corresponding principal ideals aR and bR satisfy the condition: $(aR)(bR)=aR\bigcap bR$ $(a R) (b R) = a R ⋂ b R$ . In this paper, we characterize the class of finite commutative rings with 1 for which the graph $G_P(R)$ $G_{P} (R)$ is complete. Here we are able to show that the graph $G_P(R)$ $G_{P} (R)$ is a line graph of some graph G if and only if $G_P(R)$ $G_{P} (R)$ is complete. For $n=p_1^{r_1}p_2^{r_2}\ldots p_{k}^{r_k}$ $n = p_{1}^{r_{1}} p_{2}^{r_{2}} \dots p_{k}^{r_{k}}$ , we show that chromatic number of $G_P(\mathbb {Z}_n)$ $G_{P} (Z_{n})$ is equal to the sum of the number of regular elements in $\mathbb {Z}_n$ $Z_{n}$ and the number of integers i such that ${r_{i}}>1$ $r_{i} > 1$ . Moreover, we characterize those n for which the graph $G_P(\mathbb {Z}_n)$ $G_{P} (Z_{n})$ is end-regular.

On the graph \(G_P(R)\) over commutative ring R

摘要