Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap

Details Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap

2009-04-11

arxiv.org/abs/0904.1800v2

Electron. J. Combin. Vol 16, no. 1, N29 (2009)

Mathematics

Combinatorics

Shorter version. Published in the Electron. J. of Combinatorics

Scientific paper

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.

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