Kirchhoff index for circulant graphs and its asymptotics

The aim of this paper is to find an analytical formula for the Kirchhoff index of circulant graphs $C_n(s_1,s_2,\dots,s_k)$ and $C_{2n}(s_1,s_2,\dots,s_k,n)$ with even and odd valency, respectively. The asymptotic behavior of the Kirchhoff index as $n\to\infty$ 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 $n\to\infty$.