Planarity of the bipartite graph associated to squares of elements and subgroups of a group

A new kind of bipartite graph is defined in this paper. The work is motivated mainly from a graph introduced by Al-Kaseasbeh and Erfanian [A bipartite graph associated to elements and cosets of subgroups of a finite group, AIMS Math. 6(10) (2021) 10395–10404] and partially from the undirected power graph of a group and from the inclusion graph. For a nontrivial group [Formula: see text], we define a simple undirected bipartite graph [Formula: see text] with the vertex set [Formula: see text], where [Formula: see text] is the set of all elements of the group [Formula: see text] and [Formula: see text] is the set of all subgroups [Formula: see text] of [Formula: see text] such that [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Here, we classify all the finite groups [Formula: see text] whose bipartite graphs [Formula: see text] are planar. In addition, we also classify the finite groups [Formula: see text] such that [Formula: see text] is outerplanar, maximal planar, maximal outerplanar, star, tree, claw graph, respectively. The planarity of the graph [Formula: see text] corresponding to an infinite group [Formula: see text] has also been investigated here. It is also observed that [Formula: see text] satisfies Vizing’s conjecture.