Intersection graphs of ideals of rings

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