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 Title: On k-Walk-Regular Graphs Dalfó Simó, Cristina; 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 Considering a connected graph $G$ with diameter $D$, we say that it is \emph{$k$-walk-regular}, for a given integer $k$ $(0\leq k \leq D)$, if the number of walks of length $\ell$ between vertices $u$ and $v$ only depends on the distance between them, provided that this distance does not exceed $k$. Thus, for $k=0$, this definition coincides with that of walk-regular graph, where the number of cycles of length $\ell$ rooted at a given vertex is a constant through all the graph. In the other extreme, for $k=D$, we get one of the possible definitions for a graph to be distance-regular. In this paper we present some algebraic characterizations of $k$-walk-regularity, which are based on the so-called local spectrum and predistance polynomials of $G$. Moreover, some results, concerning some parameters of a geometric nature, such as the cosines, and the spectrum of walk-regular graphs are presented. Graph theorywalk-regular graphdistance-regular graphk-walk-regular graphlocal spectrumpredistance polynomialsGrafs, Teoria deClassificació AMS::05 Combinatorics::05C Graph theory Attribution-NonCommercial-NoDerivs 2.5 Spain http://creativecommons.org/licenses/by-nc-nd/2.5/es/ Article

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