Cayley Graph on Symmetric Group Generated by Elements Fixing $k$ Points

TL;DR

This paper studies the spectral properties of a Cayley graph on the symmetric group generated by elements fixing exactly k points, deriving a recurrence for eigenvalues and analyzing their signs for the case k=1.

Contribution

It introduces a recurrence formula for the eigenvalues of the k-point fixing graph on the symmetric group and analyzes their signs for specific cases.

Findings

Derived a recurrence formula for eigenvalues of the graph

Determined the sign of eigenvalues for the case k=1

Enhanced understanding of spectral properties of symmetric group Cayley graphs

Abstract

Let $\mathcal{S}_{n}$ be the symmetric group on $[n]=\{1, \ldots, n\}$. The $k$-point fixing graph $\mathcal{F}(n,k)$ is defined to be the graph with vertex set $\mathcal{S}_{n}$ and two vertices $g$, $h$ of $\mathcal{F}(n,k)$ are joined if and only if $gh^{-1}$ fixes exactly $k$ points. In this paper, we derive a recurrence formula for the eigenvalues of $\mathcal{F}(n,k)$. Then we apply our result to determine the sign of the eigenvalues of $\mathcal{F}(n,1)$.