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 .
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 ...
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 .
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.