Strongly walk-regular graphs

We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an ℓ > 1 such that the number of walks of length ℓ 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 ℓ-walk-regular for all ℓ, whereas the graphs from the fourth family are ℓ-walk-regular for every odd ℓ. The case of regular graphs with four eigenvalues is the most interesting (and complicated) one. Such graphs cannot be strongly ℓ-walk-regular for even ℓ. We will characterize the case that regular four-eigenvalue graphs are strongly ℓ-walk-regular for every odd ℓ, 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 ℓ-walk-regular for at most one ℓ. 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 ℓ-walk-regular for only one particular ℓ different from 3.