On Sombor Index of Graphs

TL;DR

This paper investigates the Sombor index, a new graph invariant based on vertex degrees, establishing sharp bounds for connected graphs and characterizing the extremal cases.

Contribution

It provides the first sharp bounds for the Sombor index of connected graphs and characterizes the graphs that attain these bounds.

Findings

Sharp lower bounds for the Sombor index of connected graphs.

Sharp upper bounds for the Sombor index of connected graphs.

Characterization of graphs attaining the bounds.

Abstract

Recently, Gutman defined a new vertex-degree-based graph invariant, named the Sombor index $SO$ of a graph $G$, and is defined by $$SO(G)=\sum_{uv\in E(G)}\sqrt{d_G(u)^2+d_G(v)^2},$$ where $d_G(v)$ is the degree of the vertex $v$ of $G$. In this paper, we obtain the sharp lower and upper bounds on $SO(G)$ of a connected graph, and characterize graphs for which these bounds are attained.