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

TL;DR

This paper proves that Cayley graphs on symmetric groups generated by n-cycles are hyperenergetic, integral, and provides explicit formulas for their energy and nullity, expanding understanding of graph spectra in algebraic combinatorics.

Contribution

It establishes that these Cayley graphs are hyperenergetic and integral, with explicit formulas for their energy and nullity, a novel result in algebraic graph theory.

Findings

Cayley graphs generated by n-cycles are hyperenergetic for all n ≥ 4.

These graphs are integral with explicitly computed energy.

Nullity of these graphs is given by a specific combinatorial formula.

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