One more proof about the spectrum of Transposition graph

TL;DR

This paper explores the spectrum of the transposition graph $T_n$, a Cayley graph over the symmetric group, by leveraging the spectral properties of Jucys-Murphy elements, providing new insights into its eigenvalues.

Contribution

It introduces a novel method to determine the spectrum of $T_n$ using the spectral properties of Jucys-Murphy elements, connecting algebraic and combinatorial aspects.

Findings

Spectrum of $T_n$ can be derived from Jucys-Murphy elements

Provides a new algebraic approach to spectral analysis of Cayley graphs

Enhances understanding of the eigenstructure of transposition graphs

Abstract

A Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group $Sym_n$ generated by all transpositions. This paper shows how the spectrum of $T_n$ can be obtained using the spectral properties of the Jucys-Murphy elements.