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The local spectrum of a vertex set in a graph has been proven to be veryuseful to study some of its metric properties. It also has applications in thearea of pseudo-distance-regularity around a set and can be used to obtainquasi-spectral characterizations of completely (pseudo-)regular codes. In thispaper we study the relation between the local spectrum of a vertex set and thelocal spectrum of each of its subconstituents. Moreover, we obtain a newcharacterization for completely pseudo-regular codes, and consequently forcompletely regular codes, in terms of the relation between the local spectrumof an extremal set of vertices and the local spectrum of its antipodal set. Wealso present a new proof of the version of the Spectral Excess Theorem forextremal sets of vertices.

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