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

Abstract

The general sum-connectivity index of a graph $G$ is defined as $chi_{alpha}(G)= sum_{uvin E(G)} (d_u + d_{v})^{alpha}$ where $d_{u}$ is degree of the vertex $uin V(G)$, $alpha$ 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 $chi_{alpha}$ values of polyomino chain graphs) give the complete solution of the aforementioned problem.