Intersection graph of cyclic subgroups of groups

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