Prime intersection graph of ideals of a ring (Q2174838)

Let \(R\) be a ring. The prime intersection graph of ideals of \(R,\) denoted by \(G_P(R),\) is the graph whose vertex set is the collection of all non-trivial (left) ideals of \(R\) with two distinct vertices \(I\) and \(J\) are adjacent if and only if \(I \cap J\neq 0\) and either one of \(I\) or \(J\) is a prime ideal of \(R.\) In this paper, authors are interested in the properties like diameter, girth, disconnected, Eulerian, planar, complete and chromatic number for \(G_P(\mathbb{Z}_n).\) Further it is proved that \(G_P(\mathbb{Z}_n)\) is not Hamiltonian and diameter of \(G_P(\mathbb{Z}_n)\) is at most 2. In the general situtation, it is proved that \(G_P(R)\) is connected when \(R\) is an integral domain but not a field.