Some results on a subgraph of the intersection graph of ideals of a commutative ring

The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal.   Let $R$ be a ring. Let us denote  the collection  of all proper ideals  of $R$ by $mathbb{I}(R)$  and $mathbb{I}(R)backslash {(0)}$ by $mathbb{I}(R)^{*}$.  With $R$, we associate an undirected graph denoted by $g(R)$, whose vertex set is $mathbb{I}(R)^{*}$ and distinct vertices $I_{1}, I_{2}$ are adjacent in $g(R)$  if and only if $I_{1}cap I_{2}neq I_{1}I_{2}$.  The aim of this article is to study the interplay between the graph-theoretic properties of $g(R)$ and the ring-theoretic properties of $R$.