Spectrum of the Transposition graph

TL;DR

This paper investigates the eigenvalues of the transposition graph $T_n$, proving the existence of certain eigenvalues for large $n$ and providing exact values for some of the largest eigenvalues.

Contribution

It establishes the presence of all integers up to a certain point as eigenvalues of $T_n$ for large $n$, and computes specific eigenvalues with multiplicities.

Findings

Zero is an eigenvalue of $T_n$ for all $n eq 2$.

One is an eigenvalue of $T_n$ for odd $n extgreater= 7$ and even $n extgreater= 14$.

Exact values of the third and fourth largest eigenvalues are provided.

Abstract

Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group generated by all transpositions. It is known that all eigenvalues of $T_n$ are integers. However, an explicit description of the spectrum is unknown. In this paper we prove that for any integer $k\geqslant 0$ there exists $n_0$ such that for any $n\geqslant n_0$ and any $m \in \{0, \dots, k\}$, $m$ is an eigenvalue of $T_n$. In particular, it is proved that zero is an eigenvalue of $T_n$ for any $n\neq2$, and one is an eigenvalue of $T_n$ for any odd $n\geqslant 7$ and for any even $n \geqslant 14$. We also present exact values of the third and the fourth largest eigenvalues of $T_n$ with their multiplicities.