Note on Sombor index of connected graphs with given degree sequence

TL;DR

This paper investigates extremal properties of the Sombor index in connected graphs with fixed degree sequences, identifying BFS-graphs as extremal for various parameter ranges of alpha.

Contribution

It establishes the existence and uniqueness of extremal BFS-graphs with minimum or maximum Sombor index under specific conditions and parameter ranges.

Findings

BFS-graphs are extremal for the Sombor index with fixed degree sequences.

Unique extremal BFS-graphs exist for trees, unicyclic, and bicyclic graphs.

Extremal graphs minimize or maximize the Sombor index depending on alpha's value.

Abstract

For a simple connected graph $G=(V,E)$, let $d(u)$ be the degree of the vertex $u$ of $G$. The general Sombor index of $G$ is defined as $$SO_{\alpha}(G)=\sum_{uv\in E} \left[d(u)^2+d(v)^2\right]^\alpha$$ where $SO(G)=SO_{0.5}(G)$ is the recently invented Sombor index. In this paper, we show that in the class of connected graphs with a fixed degree sequence (for which the minimum degree being equal to one), there exists a special extremal $BFS$-graph with minimum general Sombor index for $0<\alpha<1$ (resp. maximum general Sombor index for either $\alpha>1$ or $\alpha<0$). Moreover, for any given tree, unicyclic, and bicyclic degree sequences with minimum degree 1, there exists a unique extremal $BFS$-graph with minimum general Sombor index for $0<\alpha<1$ and maximum general Sombor index for either $\alpha>1$ or $\alpha<0$.