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

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.

19 pages