Finite nilpotent groups with isomorphic inclusion graphs of cyclic subgroups

The inclusion graph of cyclic subgroups of a finite group [Formula: see text], [Formula: see text], is the (undirected) graph with vertex set consisting of all cyclic subgroups of [Formula: see text] and two distinct vertices are adjacent if one is a subgroup of the other. In this paper, we first prove that, if [Formula: see text] is a [Formula: see text]-group with [Formula: see text], [Formula: see text], and [Formula: see text] is a group such that [Formula: see text], then [Formula: see text] is a [Formula: see text]-group and their directed graphs are isomorphic, too. We also prove that any two nilpotent groups have the same [Formula: see text]-graphs if and only if their sylow subgroups have the same [Formula: see text]-graphs. Moreover, any two nilpotent groups have the same [Formula: see text]-graphs if and only if they have the same directed [Formula: see text]-graphs.