Spectrum of Cayley graphs on the symmetric group generated by transpositions

TL;DR

This paper investigates the spectral properties of a specific Cayley graph on the symmetric group generated by transpositions involving the first element, revealing that its spectrum includes all integers from -(n-1) to n-1, with some exceptions.

Contribution

It provides a complete description of the spectrum of Cayley graphs generated by transpositions on the symmetric group, highlighting the range of eigenvalues.

Findings

Spectrum contains all integers from -(n-1) to n-1

Zero is excluded from the spectrum for n=2 or n=3

Spectrum characterization applies to Cayley graphs generated by specific 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 $n=3$).