A bipartite graph associated to elements and cosets of subgroups of a finite group

Let $ G $ be a finite group. A bipartite graph associated to elements and cosets of subgroups of $ G $ is the simple undirected graph $ \Gamma(G) $ with the vertex set $ V(\Gamma(G)) = A\cup B $, where $ A $ is the set of all elements of a group $ G $ and $ B $ is the set of all subgroups of a group $ G $ and two vertices $ x \in A $ and $ H \in B $ are adjacent if and only if $ xH = Hx $. In this article, several graph theoretical properties are investigated. Also, we obtain the diameter, girth, and the dominating number of $ \Gamma(G) $. We discuss the planarity and outer planarity for $ \Gamma(G) $. Finally, we prove that if $ p $ and $ q $ are distinct prime numbers and $ n = pq^k $, where $ p < q $ and $ k\geq 1 $, then $ \Gamma(D_{2n}) $ is not Hamiltonian.