Abstract
The Sombor index, which Gutman proposed recently in 2021, is a graph-based degree-related topological descriptor with potential applicability in understanding how compounds behave thermodynamically. Let \(\mathbb {V}_\nu ^\rho \) (resp. \(\mathbb {E}_\nu ^\rho \)) denote the collection of all \(\nu \)-vertex connected graphs having number of cut-vertices (resp. edge-connectivity) \(\rho \). In Problem 1 of [On Sombor index of graphs with a given number of cut-vertices, MATCH Commun. Math. Comput. Chem., 89 (2023) 437–450], the authors asked to find maximum Sombor index of graph in \(\mathbb {V}_\nu ^\rho \). Moreover, in Remark 1 of [On the Sombor index of graphs with given connectivity and number of bridges, arXiv:2208.09993, (2022)], the authors raise a conjecture on the maximum Sombor index in \(\mathbb {E}_\nu ^\rho \). This paper solves both of the open problems and find sharp upper bounds on the Sombor index of graphs in \(\mathbb {V}_\nu ^\rho \) and \(\mathbb {E}_\nu ^\rho \). The respective maximum graphs achieving the bounds have also been classified.
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Acknowledgements
The authors are indebted to the anonymous reviewers’ for suggesting improvements to the initial submission of the paper.
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S. Hayat is supported by UBD Faculty Research Grants (No. UBD/RSCH/1.4/FICBF(b)/2022/053).
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Hayat, S., Arshad, M. & Gutman, I. Proofs to Some Open Problems on the Maximum Sombor Index of Graphs. Comp. Appl. Math. 42, 279 (2023). https://doi.org/10.1007/s40314-023-02423-6
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DOI: https://doi.org/10.1007/s40314-023-02423-6