On the intersection graph of ideals of $\mathbb{Z}_m$

TL;DR

This paper introduces and analyzes a new graph structure based on ideals of the ring rac{rac{Z}_m, exploring its properties and invariants, and characterizing when it is Eulerian.

Contribution

It defines a novel graph associated with rac{rac{Z}_m, studies its properties, and determines conditions for Eulerian graphs, extending the understanding of ideal intersection graphs.

Findings

Computed girth, independence, domination, maximum degree, and chromatic index.

Established conditions for the graph to be Eulerian.

Provided structural insights into the graph's properties.

Abstract

Let $m>1$ be an integer, and let $I(\mathbb{Z}_m)^*$ be the set of all non-zero proper ideals of $\mathbb{Z}_m$. The intersection graph of ideals of $\mathbb{Z}_m$, denoted by $G(\mathbb{Z}_m)$, is a graph with vertices $I(\mathbb{Z}_m)^*$ and two distinct vertices $I,J\in I(\mathbb{Z}_m)^*$ are adjacent if and only if $I\cap J\neq 0$. Let $n>1$ be an integer and $\mathbb{Z}_n$ be a $\mathbb{Z}_m$-module. In this paper, we introduce and study a kind of graph structure of $\mathbb{Z}_m$, denoted by $G_n(\mathbb{Z}_m)$. It is the undirected graph with the vertex set $I(\mathbb{Z}_m)^*$, and two distinct vertices $I$ and $J$ are adjacent if and only if $I\mathbb{Z}_n\cap J\mathbb{Z}_n\neq 0$. Clearly, $G_m(\mathbb{Z}_m)=G(\mathbb{Z}_m)$. We obtain some graph theoretical properties of $G_n(\mathbb{Z}_m)$ and we compute some of its numerical invariants, namely girth, independence number, domination number, maximum degree and chromatic index. We also determine all integer numbers $n$ and $m$ for which $G_n(\mathbb{Z}_m)$ is Eulerian.