On the module intersection graph of ideals of rings

Let R be a commutative ring and M an R-module. The M -intersection graph of ideals of R is an undirected simple graph, denoted by GM (R), whose vertices are non-zero proper ideals of R and two distinct vertices are adjacent if and only if IM ∩ JM 6= 0. In this article, we focus on how certain graph theoretic parameters of GM (R) depend on the properties of both R and M . Specifically, we derive a necessary and sufficient condition for R and M such that the M -intersection graph GM (R) is either connected or complete. Also, we classify all R-modules according to the diameter value of GM (R). Further, we characterize rings R for which GM (R) is perfect or Hamiltonian or pancyclic or planar. Moreover, we show that the graph GM (R) is weakly perfect and cograph.