Extremal Kirchhoff index in polycyclic chains

Abstract

The Kirchhoff index of graphs, introduced by Klein and Randi\'{c} in 1993, has been known useful in the study of computer science, complex network and quantum chemistry. The Kirchhoff index of a graph $G$ is defined as $Kf(G)=\sum\limits_{\{u,v\}\subseteq V(G)}\Omega_{G}(u,v)$, where $\Omega_{G}(u,v)$ denotes the resistance distance between $u$ and $v$ in $G$. In this paper, we determine the maximum (resp. minimum) $k$-polycyclic chains with respect to Kirchhoff index for $k\geq 5$, which extends the results of Yang and Klein [Comparison theorems on resistance distances and Kirchhoff indices of $S,T$-isomers, Discrete Appl. Math. 175 (2014) 87-93], Yang and Sun [Minimal hexagonal chains with respect to the Kirchhoff index, Discrete Math. 345 (2022) 113099], Sun and Yang [Extremal pentagonal chains with respect to the Kirchhoff index, Appl. Math. Comput. 437 (2023) 127534] and Ma [Extremal octagonal chains with respect to the Kirchhoff index, arXiv: 2209.10264].