General sum-connectivity index of trees and unicyclic graphs with fixed maximum degree

TL;DR

This paper determines the maximum general sum-connectivity index for trees and unicyclic graphs with fixed maximum degree within specific alpha ranges, characterizing extremal graphs and extending prior results.

Contribution

It extends existing results by identifying maximum indices and extremal graphs for trees and unicyclic graphs with fixed maximum degree for certain alpha ranges.

Findings

Maximum sum-connectivity index for trees with fixed degree and alpha range

Maximum sum-connectivity index for unicyclic graphs with fixed degree and alpha range

Characterization of extremal and second extremal graphs

Abstract

The general sum-connectivity index of a graph $G$ is defined as $\chi_\alpha(G)=\sum\limits_{uv\in E(G)} {(d(u)+d(v))^{\alpha}}$, where $d(v)$ denotes the degree of the vertex $v$ in $G$ and $\alpha$ is a real number. In this paper it is deduced the maximum value for the general sum-connectivity index of $n$-vertex trees for $-1.7036\leq \alpha <0$ and of $n$-vertex unicyclic graphs for $-1\le \alpha <0$ respectively, with fixed maximum degree $\triangle $. The corresponding extremal graphs, as well as the $n$-vertex unicyclic graphs with the second maximum general sum-connectivity index for $n\ge 4$ are characterized. This extends the corresponding results by Du, Zhou and Trinajsti\' c [arXiv:1210.5043] about sum-connectivity index.