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Characterizing (ℓ,m)-walk-regular graphs Dalfó, Cristina Fiol Mora, Miguel Ángel Garriga, Ernest Distance-regular graph Walk-regular graph Adjacency matrix Spectrum Predistance polynomial Preintersection number A graph G with diameter D and d + 1 distinct eigenvalues is said to be (ℓ, m)-walk-regular, for some integers ℓ ∈ [0, d] and m ∈ [0, D], ℓ≥ m, if the number of walks of length i ∈ [0, ℓ] between any pair of vertices at distance j ∈ [0, m] depends only on the values of i and j. In this paper, we study some algebraic and combinatorial characterizations of (ℓ, m)-walk-regularity based on the so-called predistance polynomials and the preintersection numbers. Research supported by the Ministerio de Educación y Ciencia (Spain) and the European Regional Development Fund under project MTM2008-06620-C03-01, and by the Catalan Research Council under project 2009SGR1387. 2010-12-30 info:eu-repo/semantics/acceptedVersion https://doi.org/10.1016/j.laa.2010.06.042 0024-3795 1873-1856 https://hdl.handle.net/10459.1/463253 eng info:eu-repo/grantAgreement/MICINN//MTM2008-06620-C03-01/ES/PROBLEMAS EXTREMALES Y DE OPTIMIZACION EN TEORIA DE GRAFOS Y COMBINATORIA: APLICACION AL ANALISIS Y ALGORITMOS DE REDES DE COMUNICACION/ Versió postprint del document publicat a: https://doi.org/10.1016/j.laa.2010.06.042 Linear Algebra and its Applications, 2010, vol. 433, núm. 11-12, p. 1821-1826 Linear Algebra and Its Applications cc-by-nc-nd, (c) Elsevier, 2010 Attribution-NonCommercial-NoDerivatives 4.0 International info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc-nd/4.0/ Elsevier