The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups

• DOI: 10.48550/arxiv.2402.02427
• Publication Date: 2024-02-04
• Authors: Yuxuan Li; Binzhou Xia; Sanming Zhou
• Keywords: Cayley graph

A Cayley graph on the symmetric group $S_n$ is said to have Aldous property if its strictly second largest eigenvalue (that is, smaller than degree) attained by standard representation of $S_n$. For $1\leq r < k n$, let $C(n,k;r)$ be set $k$-cycles moving every point in $\{1, \ldots, r\}$. Recently, Siemons and Zalesski [J. Algebraic Combin. 55 (2022) 989--1005] posed a conjecture which equivalent saying that for any $n \ge 5$ r<k<n$ nonnormal $\mathrm{Cay}(S_n, C(n,k;r))$ with connection has property. Solving this conjecture, we prove all these graphs except when (i) $(n, k, r) = (6, 5, 1)$ or (ii) $n$ odd, $k n-1$, $1 \le \frac{n}{2}$. Along way determine irreducible representations can achieve C(n,n-1;r))$ as well smallest graph.