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

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, w...

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Bibliographic Details
Published in:arXiv.org 2019-05
Main Authors: Huang, Sumin, Li, Shuchao
Format: Article
Language:English
Subjects:
Apexes
Asymptotic properties
Chains
Cylinder liners
Physicists
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1905.09017