Let G be a finite group. A bipartite graph associated to elements and cosets of subgroups of G is the simple undirected graph Γ(G) with the vertex set V(Γ(G)) = A [ 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 2 A and H 2 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 Γ(G). We discuss the planarity and outer planarity for Γ(G). Finally, we prove that if p and q are distinct prime numbers and n = pqk, where p < q and k ≥ 1, then Γ(D2n) is not Hamiltonian.

Keywords