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

Abstract

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

Theorems & Definitions (3)

• Theorem 1.1
• Corollary 1.2
• Theorem 2.1