Permutability graph of cyclic subgroups

Let $G$ be a group. \textit{The permutability graph of cyclic subgroups of $G$}, denoted by $Γ_c(G)$, is a graph with all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $Γ_c(G)$ 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.

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