The Sombor index of trees and unicyclic graphs with given matching number

TL;DR

This paper determines the trees and unicyclic graphs with the maximum Sombor index among those with fixed vertex count and matching number, expanding understanding of this topological index in graph theory.

Contribution

It identifies the extremal graphs with maximum Sombor index within classes of trees and unicyclic graphs constrained by matching number.

Findings

Identifies the tree with maximum Sombor index in _{n,m}

Identifies the unicyclic graph with maximum Sombor index in _{n,m}

Provides structural characterizations of extremal graphs

Abstract

In 2021, the Sombor index was introduced by Gutman, which is a new degree-based topological molecular descriptors. The Sombor index of a graph $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, where $d_G(v)$ is the degree of the vertex $v$ in $G$. Let $\mathscr{T}_{n,m}$ and $\mathscr{U}_{n,m}$ be the set of trees and unicyclic graphs on $n$ vertices with fixed matching number $m$, respectively. In this paper, the tree and the unicyclic graph with the maximum Sombor index are determined among $\mathscr{T}_{n,m}$ and $\mathscr{U}_{n,m}$, respectively.