Aldous' spectral gap property for normal Cayley graphs on symmetric groups

TL;DR

This paper investigates the spectral properties of normal Cayley graphs on symmetric groups, extending Aldous' spectral gap conjecture and identifying conditions under which the standard representation attains the second largest eigenvalue.

Contribution

The paper proves three new results on normal Cayley graphs on Sn, generalizing aspects of Aldous' spectral gap conjecture for large n.

Findings

Identifies conditions where the standard representation attains the second largest eigenvalue

Proves three new results on spectral properties of normal Cayley graphs

Extends the scope of Aldous' spectral gap conjecture to broader classes of graphs

Abstract

Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on Sn with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of Sn. In this paper, we prove three results on normal Cayley graphs on Sn possessing this property for sufficiently large n, one of which can be viewed as a generalization of the "normal" case of Aldous' spectral gap conjecture.