On the second largest eigenvalue of some Cayley graphs of the Symmetric Group

TL;DR

This paper analyzes the second largest eigenvalue of Cayley graphs of symmetric and alternating groups with specific generating sets, providing exact values and bounds that enhance understanding of their spectral properties.

Contribution

It derives exact formulas and bounds for the second largest eigenvalue of Cayley graphs on symmetric and alternating groups with various generating sets, extending previous work.

Findings

Exact eigenvalues for specific Cayley graphs with $H=C(n,n)$ and $H=C(n,n-1)$

A lower bound for the second largest eigenvalue with $H=C(n,k;r)$

Sharpness of the bound in the case $k=r+1$

Abstract

Let $S_n$ and $A_{n}$ denote the symmetric and alternating group on the set $\{1,.., n\},$ respectively. In this paper we are interested in the second largest eigenvalue $\lambda_{2}(\Gamma)$ of the Cayley graph $\Gamma=Cay(G,H)$ over $G=S_{n}$ or $A_{n}$ for certain connecting sets $H.$ Let $1<k\leq n$ and denote the set of all $k$-cycles in $S_{n}$ by $C(n,k).$ For $H=C(n,n)$ we prove that $\lambda_{2}(\Gamma)=(n-2)!$ (when $n$ is even) and $\lambda_{2}(\Gamma)=2(n-3)!$ (when $n$ is odd). Further, for $H=C(n,n-1)$ we have $\lambda_{2}( \Gamma)=3(n-3)(n-5)!$ (when $n$ is even) and $\lambda_{2}(\Gamma)=2(n-2)(n-5) !$ (when $n$ is odd). The case $H=C(n,3)$ has been considered in X. Huang and Q. Huang, The second largest eigenvalue of some Cayley graphs on alternating groups, J. Algebraic Combinatorics} 50(2019), $99-111$. Let $1\leq r<k<n$ and let $C(n,k;r) \subseteq C(n,k)$ be set of all $k$-cycles in $S_{n}$ which move all the points in the set $\{1,2,..., r\}.$ That is to say, $g=(i_{1},i_{2}... i_{k})(i_{k+1})\dots(i_{n})\in C(n,k;r)$ if and only if $\{1,2,..., r\}\subset \{i_{1},i_{2},..., i_{k}\}.$ Our main result concerns $\lambda_{2}( \Gamma)$, where $\Gamma=Cay(G,H)$ with $H=C(n,k;r)$ with $1\leq r<k<n$ when $G=S_{n}$ if $k$ is even and $G=A_{n}$ if $k$ is odd. Here we observe that $$\lambda_{2}( \Gamma)\geq (k-2)! {n-r \choose k-r} \frac{1}{n-r} \big((k-1)(n-k) - \frac{(k-r-1)(k-r)}{n-r-1}\big).$$ We show that this bound is sharp in the special case $k=r+1$ , giving $\lambda_{2}(\Gamma)=r!(n-r-1)$. The cases with $H=C(n,3;1)$ and $H=C(n,3;2)$ were considered earlier in the same paper of X. Huang and Q. Huang.