The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues

Let (Gamma ) be a distance-regular graph with diameter d and Kneser graph (K=Gamma _d), the distance-d graph of (Gamma ). We say that (Gamma ) is partially antipodal when K has fewer distinct eigenvalues than (Gamma ). In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues) and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with (d+1) distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a more general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.