On the maximal Sombor index of quasi-tree graphs

TL;DR

This paper investigates the maximum Sombor index among quasi-tree graphs, characterizing the extremal graphs and identifying the top three maximum values for given parameters.

Contribution

It determines the top three maximum Sombor indices for quasi-tree graphs and characterizes the extremal graphs, advancing understanding of graph indices in this class.

Findings

Identified the maximum Sombor index in quasi-tree graphs.

Characterized the extremal graphs achieving these maxima.

Determined the second and third maximum Sombor indices.

Abstract

The Sombor index $SO(G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d^2_G(u)+d^2_G(v)}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G = (V ,E)$ is called a quasi-tree, if there exists $u\in V (G)$ such that $G-u$ is a tree. Denote $\mathscr{Q}(n,k)$=\{$G$: $G$ is a quasi-tree graph of order $n$ with $G-u$ being a tree and $d_G(u)=k$\}. In this paper, we determined the maximum, the second maximum and the third maximum Sombor indices of all quasi-tree graphs in $\mathscr{Q}(n,k)$, respectively. Moreover, we characterized their corresponding extremal graphs, respectively.