Let [Formula: see text] be an integer, and let [Formula: see text] be the set of all non-zero proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a graph with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be an integer and [Formula: see text] be a [Formula: see text]-module. In this paper, we study a kind of graph structure of [Formula: see text], denoted by [Formula: see text]. It is the undirected graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Clearly, [Formula: see text]. Let [Formula: see text] and [Formula: see text], where [Formula: see text]’s are distinct primes, [Formula: see text]’s are positive integers, [Formula: see text]’s are non-negative integers, and [Formula: see text] for [Formula: see text] and let [Formula: see text], [Formula: see text]. The cardinality of [Formula: see text] is denoted by [Formula: see text]. Also, let [Formula: see text], [Formula: see text] and [Formula: see text] denote the independence number, the domination number and the set of all isolated vertices of [Formula: see text], respectively. We prove that [Formula: see text] and we show that if [Formula: see text] is not a null graph, then [Formula: see text] and [Formula: see text] We also compute some of its numerical invariants, namely maximum degree and chromatic index. Among other results, we determine all integer numbers [Formula: see text] and [Formula: see text] for which [Formula: see text] is Eulerian.