A note on the spectrum of the folded hypercube

Abstract

The folded hypercube $FQ_n$ is the Cayley graph $Cay(\mathbb{Z}_2^n,S)$, where $S=\{e_1,e_2,\dots, e_n\} \cup\{u=e_1+e_2+\dots+e_n\}$, $e_i = (0,\dots, 0, 1, 0,\dots, 0)$,  with $1$ at the $i$th position, $1\leq i \leq n$. In this paper, the spectrum of this graph is determined by an elementary and self contained method.  Then,  some properties of this graph are studied.