Combinatorial vs. algebraic characterizations of pseudo-distance-regularity around a set
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 c...
| Autores: | , , , |
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| Tipo de recurso: | informe técnico |
| Fecha de publicación: | 2009 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/3016 |
| Acceso en línea: | https://hdl.handle.net/2117/3016 |
| Access Level: | acceso abierto |
| Palabra clave: | Graph theory Combinatorics Pseudo-Distance-regular graph Adjacency matrix Local spectrum Orthogonal predistance polynomials Terwilliger algebras Completely regular code Grafs, Teoria de Combinacions (Matemàtica) Classificació AMS::05 Combinatorics::05C Graph theory Classificació AMS::05 Combinatorics::05E Algebraic combinatorics Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta |
| Sumario: | 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 $C$. |
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