Abstract

The aim of this paper is to find an analytical formula for the Kirchhoff index of circulant graphs \({{C}_{n}}({{s}_{1}},{{s}_{2}},\; \ldots ,\;{{s}_{k}})\) and \({{C}_{{2n}}}({{s}_{1}},{{s}_{2}},\; \ldots ,\;{{s}_{k}},n)\) with even and odd valency, respectively. The asymptotic behavior of the Kirchhoff index as n → ∞ is investigated. We proof that the Kirchhoff index of a circulant graph can be expressed as a sum of a cubic polynomial in n and a quantity that vanishes exponentially as n → ∞.

中文翻译：

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循环图的基尔霍夫指数及其渐近性

摘要

本文的目的是找到循环图\（{{C} _ {n}}（{{s} _ {1}}，{{s} _ {2}}的Kirchhoff指数的解析公式， \; \ ldots，\; {{s} _ {k}}）\）和\（{{C} _ {{2n}}}}（{{s} _ {1}}，{{s} _ { 2}}，\; \ ldots，\; {{s} _ {k}}，n）\）分别具有偶数和奇数价态。研究了基尔霍夫指数从n →∞的渐近行为。我们证明，循环图的基尔霍夫指数可以表示为n中三次多项式与n = ∞时指数消失的量之和。