Intersection graphs of cyclic subgroups of groups

Let G be a group. The intersection graph of cyclic subgroups of G  , denoted by Ic(G)Ic(G), is a graph having all the proper cyclic subgroups of G   as its vertices and two distinct vertices in Ic(G)Ic(G) are adjacent if and only if their intersection is non-trivial. In this paper, we classify the finite groups whose intersection graphs of cyclic subgroups are one of totally disconnected, complete, star, path, cycle. We show that for a given finite group G  , girth(Ic(G))∈{3,∞}girth(Ic(G))∈{3,∞}. Moreover, we classify all finite non-cyclic abelian groups whose intersection graphs of cyclic subgroups are planar. Also for any group G  , we determine the independence number, clique cover number of Ic(G)Ic(G) and show that Ic(G)Ic(G) is weakly α  -perfect. Among the other results, we determine the values of n for which Ic(Zn)Ic(Zn) is regular and estimate its domination number.