The Kirchhoff Index of Enhanced Hypercubes

Abstract

Let $\{e_{1},\ldots,e_{n}\}$ be the standard basis of abelian group $Z_{2}^{n}$, which can be also viewed as a linear space of dimension $n$ over the Galois filed $F_{2}$, and $\epsilon_{k}=e_k+e_{k+1}+\cdots+e_n$ for some $1\le k\le n-1$. It is well known that the so called enhanced hypercube $Q_{n, k}(1\le k \le n-1)$ is just the Cayley graph $Cay(Z_{2}^{n},S)$ where $S=\{e_{1},\ldots, e_{n},\epsilon_{k}\}$. In this paper, we obtain the spectrum of $Q_{n, k}$, from which we give an exact formula of the Kirchhoff index of the enhanced hypercube $Q_{n, k}$. Furthermore, we prove that, for a given $n$, $Kf(Q_{n, k})$ is increased with the increase of $k$. Finally, we get $\lim\limits_{n\to\infty}\frac{Kf(Q_{n, k})}{\frac{2^{2n}}{n+1}}=1$.