  Let $G$ be a group. \textit{The permutability graph of cyclic subgroups of
$G$}, denoted by $\Gamma_c(G)$, is a graph with all the proper cyclic subgroups
of $G$ as its vertices and two distinct vertices in $\Gamma_c(G)$ are adjacent
if and only if the corresponding subgroups permute in $G$. In this paper, we
classify the finite groups whose permutability graph of cyclic subgroups
belongs to one of the following: bipartite, tree, star graph, triangle-free,
complete bipartite, $P_n$, $C_n$, $K_4$, $K_{1,3}$-free, unicyclic. We classify
abelian groups whose permutability graph of cyclic subgroups are planar. Also
we investigate the connectedness, diameter, girth, totally disconnectedness,
completeness and regularity of these graphs.
