On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions

TL;DR

This paper derives explicit eigenvalue formulas for Laplacians of Cayley graphs generated by complete multipartite transpositions, confirming Aldous's conjecture for these graphs by linking spectral gaps of related processes.

Contribution

It provides a new explicit spectral characterization of Cayley graphs on symmetric groups generated by complete multipartite transpositions, confirming Aldous's conjecture in this case.

Findings

Eigenvalues expressed via irreducible characters and Littlewood-Richardson coefficients

Laplacians of the graph and Cayley graph share the same first nontrivial eigenvalue

Aldous's conjecture holds for complete multipartite graphs

Abstract

Given a finite simple graph $\cG$ with $n$ vertices, we can construct the Cayley graph on the symmetric group $S_n$ 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.