On the distance spectrum of distance regular graphs

The distance matrix of a simple graph G is D(G) = (d(ij)) where d(ij) is the distance between ith and jth vertices of G. The spectrum of the distance matrix is known as the distance spectrum or D-spectrum of G. A simple connected graph G is called distance regular if it is regular, and if for any two vertices x, y is an element of G at distance i, there are constant number of neighbors c(i) and b(i) of y at distance i - 1 and i + 1 from x respectively. In this paper we prove that distance regular graphs with diameter d have at most d + 1 distinct D-eigenvalues. We find an equitable partition and the corresponding quotient matrix of the distance regular graph which gives the distinct D-eigenvalues. We also prove that distance regular graphs satisfying b(i) = c(d-1) have at most [d/2] + 2 distinct D-eigenvalues. Applying these results we find the distance spectrum of some distance regular graphs including the well known Johnson graphs. Finally we also answer the questions asked by Lin et al. [16]. (C) 2015 Elsevier Inc. All rights reserved.