Distance-regular graphs with small number of distinct distance eigenvalues

Abstract

In this paper we characterize distance-regular graphs with diameter three having exactly three distinct distance eigenvalues, and also bipartite distance-regular graphs with diameter four having three distinct distance eigenvalues. We derive some properties and give particular examples of such graphs. We also present an infinite family of bipartite semiregular graphs with diameter four having exactly four distinct distance eigenvalues. With these results, we address some problems posed in Atik and Panigrahi (2015) [3].