On a bipartite graph defined on groups

Let G be a group and L(G) be the set of all subgroups of G. We introduce a bipartite graph & Bernoullis;(G) on G whose vertex set is the union of G x G and L(G), and the vertices (a,b) is an element of G x G and H is an element of L(G) are adjacent if H is generated by a and b. In this paper, we establish connections between & Bernoullis;(G) and the generating graph of G. 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 & Bernoullis;(G). We obtain relations between & Bernoullis;(G) and certain probabilities associated to finite groups. We also obtain expressions for various topological indices of & Bernoullis;(G). Finally, we realize the structures of & Bernoullis;(G) for the dihedral groups of order 2p and 2p(2) and dicyclic groups of order 4p and 4p(2) (where p is any prime) including certain other small order groups.