Combinatorial vs. algebraic characterizations of pseudo-distance-regularity around a set

View/Open
Document typeExternal research report
Defense date2009-06
Rights accessOpen Access
Abstract
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.
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. In the way, we also
obtain some 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 $C$.
Files | Description | Size | Format | View |
---|---|---|---|---|
terwilliger(18june).pdf | 213,6Kb | View/Open |
Except where otherwise noted, content on this work
is licensed under a Creative Commons license
:
Attribution-NonCommercial-NoDerivs 3.0 Spain