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

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.