Spectrum of Cayley graphs on the symmetric group generated by transpositions

Abstract

For an integer $n\geq 2$, let $X_n$ be the Cayley graph on the symmetric group $S_n$ generated by the set of transpositions ${(1 2),(1 3),...,(1 n)}$. It is shown that the spectrum of $X_n$ contains all integers from $-(n-1)$ to $n-1$ (except 0 if $n=2$ or $n=3$).

We have been informed by Guillaume Chapuy, Valentin Féray and Paul Renteln, that the problem in question can be solved by exploiting certain properties of the Jucys-Murphy elements, discovered by Jucys and independently by Flatto, Odlyzko and Wales