  In a recent paper Gunnells, Scott and Walden have determined the complete
spectrum of the Schreier graph on the symmetric group corresponding to the
Young subgroup $S_{n-2}\times S_2$ and generated by initial reversals. In
particular they find that the first nonzero eigenvalue, or spectral gap, of the
Laplacian is always 1, and report that "empirical evidence" suggests that this
also holds for the corresponding Cayley graph. We provide a simple proof of
this last assertion, based on the decomposition of the Laplacian of Cayley
graphs, into a direct sum of irreducible representation matrices of the
symmetric group.
