On the second eigenvalue of a Cayley graph of the symmetric group

2021/08/31 by Maleki, Roghayeh, Razafimahatratra, Andriaherimanana Sarobidy
#05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.13585

Abstract

In 2020, Siemons and Zalesski [On the second eigenvalue of some Cayley graphs of the symmetric group. \it arXiv preprint arXiv:2012.12460, 2020] determined the second eigenvalue of the Cayley graph Γ_n,k = Cay(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≥ 3 and k∈ ℕ relatively small compared to n, the second eigenvalue of Γ_n,k is the eigenvalue afforded by the irreducible character of Sym(n) that corresponds to the partition [n-1,1]. As a byproduct of our method, the result of Siemons and Zalesski when k ∈ \0,1\ is retrieved. Moreover, we prove that the second eigenvalue of Γ_n,n-5 is also equal to the eigenvalue afforded by the irreducible character of the partition [n-1,1].

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