The Spectra of Arrangement Graphs

TL;DR

This paper provides a new representation-theoretic approach to explicitly determine the spectrum of arrangement graphs' adjacency matrices, confirming a prior conjecture and simplifying existing proofs.

Contribution

It introduces a novel method using symmetric group representation theory to derive the spectrum of arrangement graphs, offering a simpler proof and confirming a previous conjecture.

Findings

Explicit spectrum formula in terms of symmetric group characters

Simplified proof of the spectrum using representation theory

Confirmation of a conjecture by Chen, Ghorbani, and Wong

Abstract

Arrangement graphs were introduced for their connection to computational networks and have since generated considerable interest in the literature. In a pair of recent articles by Chen, Ghorbani and Wong, the eigenvalues for the adjacency matrix of an (n,k)-arrangement graph are studied and shown to be integers. In this manuscript, we consider the adjaceny matrix directly in terms of the representation theory for the symmetric group. Our point of view yields a simple proof for an explicit fomula of the associated spectrum in terms of the characters of irreducibile representations evaluated on transpositions. As an application we prove a conjecture raised by Chen, Ghorbani and Wong.