The spectra of arrangement graphs

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 adjacency matrix directly in terms of the representation theory for the symmetric group. Our point of view yields a simple proof for an explicit formula of the associated spectrum in terms of the characters of irreducible representations evaluated on a transposition. As an application we prove a conjecture raised by Chen, Ghorbani and Wong.