On a bipartite graph defined on groups

Let [Formula: see text] be a group and [Formula: see text] be the set of all subgroups of [Formula: see text]. We introduce a bipartite graph [Formula: see text] on [Formula: see text] whose vertex set is the union of [Formula: see text] and [Formula: see text], and the vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] is generated by [Formula: see text] and [Formula: see text]. In this paper, we establish connections between [Formula: see text] and the generating graph of [Formula: see text]. We also discuss about various graph parameters such as independence number, domination number, girth, diameter, matching number, clique number, irredundance number, domatic number and minimum size of a vertex cover of [Formula: see text]. We obtain relations between [Formula: see text] and certain probabilities associated to finite groups. We also obtain expressions for various topological indices of [Formula: see text]. Finally, we realize the structures of [Formula: see text] for the dihedral groups of order [Formula: see text] and [Formula: see text] and dicyclic groups of order [Formula: see text] and [Formula: see text] (where [Formula: see text] is any prime) including certain other small order groups.