Intersection graphs of ideals of rings

In this paper, we consider the intersection graph G(R)G(R) of nontrivial left ideals of a ring RR. We characterize the rings RR for which the graph G(R)G(R) is connected and obtain several necessary and sufficient conditions on a ring RR such that G(R)G(R) is complete. For a commutative ring RR with identity, we show that G(R)G(R) is complete if and only if G(R[x])G(R[x]) is also so. In particular, we determine the values of nn for which G(Zn)G(Zn) is connected, complete, bipartite, planar or has a cycle. Next, we characterize finite graphs which arise as the intersection graphs of ZnZn and determine the set of all non-isomorphic graphs of ZnZn for a given number of vertices. We also determine the values of nn for which the graph of ZnZn is Eulerian and Hamiltonian.