The Kirchhoff Index of Enhanced Hypercubes

TL;DR

This paper derives the spectrum and Kirchhoff index of enhanced hypercubes, showing how the index varies with parameters and establishing its asymptotic behavior as the dimension grows.

Contribution

It provides an exact formula for the Kirchhoff index of enhanced hypercubes and analyzes its dependence on the parameter k.

Findings

Kirchhoff index increases with k for fixed n.

Exact spectral formula for enhanced hypercubes.

Asymptotic limit of Kirchhoff index ratio as n approaches infinity.

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 $\lim\limits_{n\to\infty}\frac{Kf(Q_{n, k})}{\frac{2^{2n}}{n+1}}=1$.