On the distance spectrum of certain distance biregular graphs

Abstract

In this article we present an infinite family of bipartite distance biregular graphs having an arbitrarily large diameter and whose distance matrices have exactly four distinct eigenvalues. This result answers a question posed by F. Atik and P. Panigrahi in On the distance spectrum of distance regular graphs (Linear Algebra and its Applications, 478 (2015), pp. 256-273) about the existence of connected graphs with diameter \(d\) that are not distance regular, whose distance matrix has less than \(d + 1\) distinct eigenvalues.