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 ≤ k ≤ D), if the number of walks of length l 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 l 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. Moreover, some results concerning some parameters of a geometric nature, such as the cosines, and the spectrum of walk-regular graphs are presented.