On the Second Eigenvalue of Certain Cayley Graphs on the Symmetric Group

Abstract

In 2020, Siemons and Zalesski determined the second eigenvalue of the Cayley graph $\Gamma _{n,k} = {{\,\textrm{Cay}\,}}({{\,\textrm{Sym}\,}}(n), C(n,k))$ for $k = 0$ and $k=1$, where C(n, k) is the conjugacy class of $(n-k)$-cycles. In this paper, it is proved that for any $n\ge 3$ and $k\in {\mathbb {N}}$ relatively small compared to n, the second eigenvalue of $\Gamma _{n,k}$ is the eigenvalue afforded by the irreducible character of ${{\,\textrm{Sym}\,}}(n)$ that corresponds to the partition $[n-1,1]$. As a byproduct of our method, the result of Siemons and Zalesski when $k \in \{0,1\}$ is retrieved. Moreover, we prove that the second eigenvalue of $\Gamma _{n,n-5}$ is also equal to the eigenvalue afforded by the irreducible character of the partition $[n-1,1]$.