Title: 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.