Finite nilpotent groups with isomorphic inclusion graphs of cyclic subgroups

The inclusion graph of cyclic subgroups of a finite group G, I-c(G), is the (undirected) graph with vertex set consisting of all cyclic subgroups of G and two distinct vertices are adjacent if one is a subgroup of the other. In this paper, we first prove that, if G is a p-group with exp(G) = p(alpha), alpha >= 2, and H is a group such that I-c(G)congruent to I-c(H), then H is a q-group and their directed graphs are isomorphic, too. We also prove that any two nilpotent groups have the same I-c-graphs if and only if their sylow subgroups have the same I-c-graphs. Moreover, any two nilpotent groups have the same I-c-graphs if and only if they have the same directed I-c-graphs.