Proof of a conjecture on eigenvalues of transposition graph

TL;DR

This paper proves a conjecture that all integers within a specific interval are eigenvalues of the transposition graph on symmetric groups, expanding understanding of its spectral properties.

Contribution

It establishes that every integer in a certain interval is an eigenvalue of the transposition graph, confirming a recent conjecture by Kravchuk.

Findings

All integers in the specified interval are eigenvalues.

The result confirms the conjecture by Kravchuk.

Provides spectral characterization of the transposition graph.

Abstract

The transposition graph $Cay(S_n,T_n)$ is the Cayley graph on the symmetric group $S_n$ generated by the set $T_n$ of all transpositions. In this paper, we show that each integer in the interval $\left[-{\lfloor(2n+1)/3 \rfloor\choose 2}, {\lfloor(2n+1)/3 \rfloor\choose 2}\right]$ is an eigenvalue of $Cay(S_n,T_n)$. This proves a recent conjecture by Kravchuk \cite{Kravchuk}.