Proofs to Some Open Problems on the Maximum Sombor Index of Graphs

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.