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On the algebraic theory of pseudo-distance-regulariry around a set
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada IV; Universitat Politècnica de Catalunya. COMBGRAF - Combinatòria, Teoria de Grafs i Aplicacions
Let C be a connected graph with vertex set V, adjacency matrix A, positive eigenvector and corresponding eigenvalue 0. A natural generalization of distance-regularity around a vertex subset C V , which makes sense even with non-regular graphs, is studied. This new concept is called pseudo-distance-regularity, and its definition is based on giving to each vertex u 2 V a weight which equals the corresponding entry u of and “regularizes” the graph. This approach reveals a kind of central symmetry which, in fact, is an inherent property of all kinds of distance-regularity, because of the distance partition of V they come from. We come across such a concept via an orthogonal sequence of polynomials, constructed from the “local spectrum” of C, called the adjacency polynomials because their definition strongly relies on the adjacency matrix A. In particular, it is shown that C is “tight” (that is, the corresponding adjacency polynomials attain their maxima at 0) if and only if C is pseudo-distance-regular around C. As an application, some new spectral characterizations of distance-regularity around a set and completely regular codes are given.
Peer Reviewed
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
Graph theory
Grafs, Teoria de
05C Graph theory
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
info:eu-repo/semantics/publishedVersion
Article
Elsevier
         

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