On the distance spectrum of distance regular graphs

Abstract

The distance matrix of a simple graph G is D(G)=(dij), where dij 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∈G at distance i, there are constant number of neighbors ci and bi 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 bi=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].