The Geometry of t-Spreads in k-Walk-Regular Graphs

A graph is walk-regular if the number of closed walks of length rooted at a given vertex is a constant through all the vertices for all l. For a walk-regular graph G with d+1 different eigenvalues and spectrally maximum diameter D = d, we study the geometry of its d-spreads, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a simplex (or tetrahedron in a three-dimensional case) and we compute its parameters. Moreover, the results are generalized to the case of k-walk-regular graphs, a family which includes both walk-regular and distance-regular graphs, and their t-spreads or vertices at distance t from each other. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 64: 312-322, 2010