Cayley graphs on symmetric groups generated by $n$-cycles are hyperenergetic

Abstract

Let $\Gamma$ be a simple graph with $n$ vertices. The energy of $\Gamma$, denoted by $\mathcal{E}(\Gamma )$, is defined as the sum of the absolute values of the eigenvalues of $\Gamma$. The graph $\Gamma$ is said to be hyperenergetic if $\mathcal{E}(\Gamma )>2n-2$. For the graph $\Gamma$, the multiplicity of the eigenvalue $0$, denoted by $\eta (\Gamma )$, is called the nullity of $\Gamma$. In this paper, we show that for every positive integer $n\ge 4$, the Cayley graph ${\Gamma }_n$ on the symmetric group $\mathrm{Sym}(n)$ generated by $n$-cycles is an integral hyperenergetic graph with $\mathcal{E}({\Gamma }_n)=2^{n-1}(n-1)!$ and $\eta ({\Gamma }_n)=n!-\left(\genfrac{}{}{0ex}{}{2n-2}{n-1}\right)$.

arXiv commentarXiv admin note: text overlap with arXiv:1808.06097 by other authors