Strongly walk - regular grapsh

We study a generalization of strongly regular graphs. We call a graph strongly walkregular if there is an ` > 1 such that the number of walks of length ` from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint union of complete bipartite graphs of the same size and isolated vertices, or a regular graph with four eigenvalues. Graphs from the first three families in this list are indeed strongly `-walk-regular for all `, whereas the graphs from the fourth family are `-walk-regular for every odd `. The case of regular graphs with four eigenvalues is the most interesting (and complicated) one. Such graphs cannot be strongly `-walk-regular for even `. We will characterize the case that regular four-eigenvalue graphs are strongly `-walk-regular for every odd `, in terms of the eigenvalues. There are several examples of infinite families of such graphs. We will show that every other regular four-eigenvalue graph can be strongly `-walk-regular for at most one `. There are several examples of infinite families of such graphs that are strongly 3-walk-regular. It however remains open whether there are any graphs that are strongly `-walk-regular for only one particular ` different from 3.