Zagreb indices of subgroup generating bipartite graph
Shrabani Das, Ahmad Erfanian, Rajat Kanti Nath
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Source: Crossref
Published: Oct 22, 2025
DOI: 10.1142/s1793830925501411
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Let [Formula: see text] be a group and [Formula: see text] be the set of all subgroups of [Formula: see text]. The subgroup generating bipartite graph [Formula: see text] defined on [Formula: see text] is a bipartite graph whose vertex set is partitioned into two sets [Formula: see text] and [Formula: see text], and two 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 deduce expressions for first and second Zagreb indices of [Formula: see text] and obtain a condition such that [Formula: see text] satisfy [Formula: see text]Hansen–Vukičević conjecture [P. Hansen and D. Vukičević, Comparing the Zagreb indices, Croat. Chem. Acta 80(2) (2007) 165–168]. It is shown that [Formula: see text] satisfies Hansen–Vukičević conjecture if [Formula: see text] is a cyclic group of order [Formula: see text], [Formula: see text] and [Formula: see text]; dihedral group of order [Formula: see text] and [Formula: see text]; and dicyclic group of order [Formula: see text] and [Formula: see text] for any prime [Formula: see text]. While computing Zagreb indices of [Formula: see text] we have computed [Formula: see text] for all [Formula: see text] 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 [Formula: see text].
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