On the general Sombor index of unicyclic graphs with a given diameter

TL;DR

This paper establishes an upper bound for the general Sombor index of unicyclic graphs with a fixed diameter for 0<α<1, and characterizes the extremal graph achieving this bound.

Contribution

It provides the first known upper bound for the general Sombor index in unicyclic graphs with fixed diameter and identifies the extremal graph.

Findings

Derived an upper bound for SO_α(G) for unicyclic graphs

Characterized the extremal unicyclic graph achieving the bound

Focused on the case 0<α<1

Abstract

The general Sombor index of $G$ is defined as $SO_{\alpha}(G)= \sum_{uv\in G}\left(d^2_{G}(u)+d^2_{G}(v)\right)^{\alpha}$. For $0<\alpha<1$, we have the upper bound of $SO_{\alpha}(G)$ on unicyclic graphs with a fixed diameter, and the extremal graph is also characterized.