A note on polyomino chains with extremum general sum-connectivity index

TL;DR

This paper characterizes polyomino chain graphs with extremum general sum-connectivity index for negative alpha, completing the understanding of their extremal properties in this class.

Contribution

It provides a complete characterization of extremal polyomino chain graphs with respect to the general sum-connectivity index for alpha<0.

Findings

Identifies graphs with maximum and minimum sum-connectivity index for alpha<0.

Completes the classification of extremal polyomino chains for all alpha values.

Extends previous results to the case of negative alpha.

Abstract

The general sum-connectivity index of a graph $G$ is defined as $\chi_{\alpha}(G)= \sum_{uv\in E(G)} (d_u + d_{v})^{\alpha}$ where $d_{u}$ is degree of the vertex $u\in V(G)$, $\alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $\chi_{\alpha}$ values from a certain collection of polyomino chain graphs is solved for $\alpha<0$. The obtained results together with already known results (concerning extremum values of polyomino chain graphs) give the complete solution of the aforementioned problem.