Let ???? be a group and ????⁡(????) be the set of all subgroups of ????. The subgroup generating bipartite graph mi(????) defined on ???? is a bipartite graph whose vertex set is partitioned into two sets ????×???? and ????⁡(????), and two vertices (????,????)∈????×???? and ????∈????⁡(????) are adjacent if ???? is generated by ???? and ????. In this paper, we deduce expressions for first and second Zagreb indices of mi(????) and obtain a condition such that mi(????) satisfy Hansen–Vukičević conjecture [P. Hansen and D. Vukičević, Comparing the Zagreb indices, Croat. Chem. Acta80(2) (2007) 165–168]. It is shown that mi(????) satisfies Hansen–Vukičević conjecture if ???? is a cyclic group of order 2⁢????,2⁢????2,4⁢????, 4⁢????2 and ????????; dihedral group of order 2⁢???? and 2⁢????2; and dicyclic group of order 4⁢???? and 4⁢????2 for any prime ????. While computing Zagreb indices of mi(????) we have computed degmi(????)⁡(????) for all ????∈????⁡(????) for the above-mentioned groups. Using these information we also compute Randic Connectivity index, Atom-Bond Connectivity index, Geometric–Arithmetic index, Harmonic index and Sum-Connectivity index of mi(????).

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