On $k$-Walk-Regular Graphs
Considering a connected graph $G$ with diameter $D$, we say that it is $k$-walk-regular, for a given integer $k$ $(0\leq k \leq D)$, if the number of walks of length $\ell$ between any pair of vertices 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 show some algebraic characterizations of $k$-walk-regularity, which are based on the so-called local spectrum and predistance polynomials of $G$.