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

Abstract

Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group S n with respect to a set of transpositions is achieved by the standard representation of S n . 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 S n with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of S n . In this paper, we prove three results on normal Cayley graphs on S n 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.