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

Abstract

Let $m>1$ be an integer, and let $I({\mathbb{Z}}_m{)}^{\ast }$ 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 $I({\mathbb{Z}}_m{)}^{\ast }$ and two distinct vertices $I,J\in I({\mathbb{Z}}_m{)}^{\ast }$ are adjacent if and only if $I\cap J\ne 0$. 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 $I({\mathbb{Z}}_m{)}^{\ast }$, and two distinct vertices $I$ and $J$ are adjacent if and only if $I{\mathbb{Z}}_n\cap J{\mathbb{Z}}_n\ne 0$. 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.

Cited by in corpus (1)

• Planar, outerplanar and ring graph of the intersection graph