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 $n\geq 4$, 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 $\eta(\Gamma_n)=n!-\binom{2n-2}{n-1}$.