TAILIEUCHUNG - Báo cáo toán học: "vs. Algebraic Characterizations of Completely Pseudo-Regular Codes"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: vs. Algebraic Characterizations of Completely Pseudo-Regular Codes. | Combinatorial vs. Algebraic Characterizations of Completely Pseudo-Regular Codes M. Camara J. Fabrega . Fiol and E. Garriga Departament de Matematica Aplicada IV Universitat Politecnica de Catalunya Jordi Girona 1-3 Mòdul C3 Campus Nord 08034 Barcelona Catalonia Spain mcamara j f abrega fiol egarriga @ Submitted Nov 16 2009 Accepted Feb 25 2010 Published Mar 8 2010 Mathematics Subject Classification 05C50 05E30 Abstract Given a simple connected graph r and a subset of its vertices C the pseudodistance-regularity around C generalizes for not necessarily regular graphs the notion of completely regular code. We then say that C is a completely pseudoregular 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 r 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 r with an endvertex in C. 1 Preliminaries Pseudo-distance-regularity is a natural generalization of distance-regularity which extends this notion to not necessarily regular graphs. The key point of this generalization relays Research supported by the Ministerio de Ciencia e Innovacion Spain with the European Regional Development Fund under projects MTM2008-06620-C03-01 and by the Catalan Research Council under project 2005SGR00256. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R37 1 on defining an adequate weight for each vertex in such a way that we obtain a regularized graph. Since its introduction in 7 the .

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