85 research outputs found
On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions
Given a finite simple graph \cG with vertices, we can construct the
Cayley graph on the symmetric group generated by the edges of \cG,
interpreted as transpositions. We show that, if \cG is complete multipartite,
the eigenvalues of the Laplacian of \Cay(\cG) have a simple expression in
terms of the irreducible characters of transpositions, and of the
Littlewood-Richardson coefficients. As a consequence we can prove that the
Laplacians of \cG and of \Cay(\cG) have the same first nontrivial
eigenvalue. This is equivalent to saying that Aldous's conjecture, asserting
that the random walk and the interchange process have the same spectral gap,
holds for complete multipartite graphs.Comment: 29 pages. Includes modification which appear on the published version
in J. Algebraic Combi
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