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

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