The Kirchhoff Index of Enhanced Hypercubes

Abstract

Let $\{e_1,\dots ,e_n\}$ 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 $S=\{e_1,\dots ,e_n,{\epsilon }_k\}$. 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 $\underset{n\to \infty }{\lim }\frac{Kf(Q_{n,k})}{\frac{2^{2n}}{n+1}}=1$.