Planar, outerplanar and ring graph of the intersection graph

TL;DR

This paper investigates the conditions under which the intersection graph of non-zero proper ideals of a modular ring, viewed as a graph, is planar, outerplanar, or a ring graph, based on parameters m and n.

Contribution

It characterizes the specific values of m and n for which the intersection graph of ideals is planar, outerplanar, or a ring graph.

Findings

Identifies when the intersection graph is planar.

Determines conditions for outerplanarity.

Classifies when the graph forms a ring graph.

Abstract

Let $m, n > 1$ be two integers, and $\mathbb{Z}_n$ be a $\mathbb{Z}_m$-module. Let $I(\mathbb{Z}_m)^*$ be the set of all non- zero proper ideals of $\mathbb{Z}_m$. The $\mathbb{Z}_n$-intersection graph of $\mathbb{Z}_m$, denoted by $G_n(\mathbb{Z}_m)$ is a graph with the vertex set $I(\mathbb{Z}_m)^*$, and two distinct vertices $I$ and $J$ are adjacent if and only if $I\mathbb{Z}_n\cap J\mathbb{Z}_n\neq 0$. In this paper, we determine the values of $m$ and $n$ for which $G_n(\mathbb{Z}_m)$ is planar, outerplanar or ring graph.