On the local spectra of the subconstituents of a vertex set and completely pseudo-regular codes

In this paper we study the relation between the local spectrum of a vertex set CC and the local spectra of its subconstituents. In particular, it is shown that, when CC is a completely regular code, such spectra are uniquely determined by the local spectra of CC. Moreover, we obtain a new characterization for completely pseudo-regular codes, and consequently for completely regular codes, in terms of the relation between the local spectrum of an extremal set of vertices and the local spectrum of its antipodal set. We also present a new proof of the version of the spectral excess theorem for extremal sets of vertices.