  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.
