The spectrum of eigenvalues for certain subgraphs of the k-point fixing graph

Abstract

Let ${\mathcal{S}}_n$ be the symmetric group on n-points. 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 $g{h}^{-1}$ fixes exactly k points. In this paper, we give a recurrence formula for the eigenvalues of a class of regular subgraphs of $\mathcal{F}(n,k)$. By using this recurrence formula, we will determine the smallest eigenvalues for this class of regular subgraphs of $\mathcal{F}(n,1)$ for sufficiently large n.