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{\bf M. C\'amara, J. F\`abrega, M.A. Fiol and E. Garriga}
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{\bf Combinatorial vs.\ Algebraic Characterizations of Completely Pseudo-Regular Codes}
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Given a simple connected graph $\Gamma$ and a subset of its vertices
$C$, the pseudo-distance-regularity around $C$ generalizes, for not
necessarily regular graphs, the notion of completely regular code.
We then say that $C$ is a completely pseudo-regular code. Up to now,
most of the characterizations of pseudo-distance-regularity has been
derived from a combinatorial definition. In this paper we propose an
algebraic (Terwilliger-like) approach to this notion, showing its
equivalence with the combinatorial one. This allows us to give new
proofs of known results, and also to obtain new characterizations
which do not depend on the so-called $C$-spectrum of $\Gamma$, but
only on the positive eigenvector of its adjacency matrix. Along the
way, we also obtain some new results relating the local spectra of a
vertex set and its antipodal. As a consequence of our study, we
obtain a new characterization of a completely regular code $C$, in
terms of the number of walks in $\Gamma$ with an endvertex in $C$.



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