On the resistance distance and Kirchhoff index of a linear hexagonal (cylinder) chain

Abstract

The resistance between two nodes in some resistor networks has been studied extensively by mathematicians and physicists. Let $L_n$ be a linear hexagonal chain with $n$\, 6-cycles. Then identifying the opposite lateral edges of $L_n$ in ordered way yields the linear hexagonal cylinder chain, written as $R_n$. We obtain explicit formulae for the resistance distance $r_{L_n}(i, j)$ (resp. $r_{R_n}(i,j)$) between any two vertices $i$ and $j$ of $L_n$ (resp. $R_n$). To the best of our knowledge $\{L_n\}_{n=1}^{\infty}$ and $\{R_n\}_{n=1}^{\infty}$ are two nontrivial families with diameter going to $\infty$ for which all resistance distances have been explicitly calculated. We determine the maximum and the minimum resistance distances in $L_n$ (resp. $R_n$). The monotonicity and some asymptotic properties of resistance distances in $L_n$ and $R_n$ are given. As well we give formulae for the Kirchhoff indices of $L_n$ and $R_n$ respectively.