On the distance spectrum of distance regular graphs

The distance matrix of a simple graph G   is D(G)=(dij)D(G)=(dij), where dijdij 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∈Gx,y∈G at distance i  , there are constant number of neighbors cici and bibi of y   at distance i−1i−1 and i+1i+1 from x respectively. In this paper we prove that distance regular graphs with diameter d   have at most d+1d+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 bi=cd−1bi=cd−1 have at most ⌈d2⌉+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].