Maximal polyomino chains with respect to the Kirchhoff index

Abstract

Let $G$ be a connected graph. The resistance distance between two vertices $u$ and $v$ of $G$ is defined as the potential difference generated between $u$ and $v$ induced by the unique $u\to v$ flow when a unit current flows in from node $u$ and flows out from node $v$. The Kirchhoff index of $G$ is defined as the sum of all the resistance distances pairs of $G$. Polyomino chains, as an important geometric structure, have been widely studied in statistical physics and mathematical chemistry. In this paper, by employing standard techniques from electrical networks and using comparison results on the Kirchhoff index of $S,T$-isomers, we first show that among all polyomino chains with $n$ squares, the maximum Kirchhoff index is attained only when the polyomino chain is a “bend-free” chain. Furthermore, according to the recursion formula for the resistance distances, “bend-free” chains with maximum and minimum Kirchhoff index are characterized. As a result, the polyomino chains with maximum Kirchhoff index are obtained.