On the Sombor index of graphs with given connectivity and number of bridges

TL;DR

This paper investigates the extremal values of the Sombor index for graphs with fixed connectivity and bridges, providing characterizations for graphs with minimum and maximum Sombor index within these classes.

Contribution

It introduces new characterizations of graphs with extremal Sombor indices based on connectivity and bridges, expanding understanding of this index's behavior.

Findings

Graphs with minimum Sombor index in bridge classes characterized.

Graphs with maximum Sombor index in vertex-connected classes characterized.

New auxiliary operations on graphs developed for these characterizations.

Abstract

Recently in 2021, Gutman introduced the Sombor index of a graph, a novel degree-based topological index. It has been shown that the Sombor index efficiently models the thermodynamic properties of chemical compounds. Assume $\mathbb{B}_n^k$ (resp. $\mathbb{V}_n^k$) comprises all graphs with order $n$ having number of bridges (resp. vertex-connectivity) $k$. Horoldagva & Xu (2021) characterized graphs achieving the maximum Sombor index of graphs in $\mathbb{B}_n^k$. This paper characterizes graphs achieving the minimum Sombor index in $\mathbb{B}_n^k$. Certain auxiliary operation on graphs in $\mathbb{B}_n^k$ are introduced and employed for the characterization. Moreover, we characterize graphs achieving maximum Sombor index in $\mathbb{V}_n^k$. ome open problems, which naturally arise from this work, have been proposed at the end.