On k-Walk-Regular Graphs

Abstract

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.