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

The local spectrum of a vertex set in a graph has been proven to be very useful to study some of its metric properties. It also has applications in the area of pseudo-distance-regularity around a set and can be used to obtain quasi-spectral characterizations of completely (pseudo-)regular codes. In this paper we study the relation between the local spectrum of a vertex set and the local spectrum of each of its subconstituents. 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.