Local WL invariance and hidden shades of regularity

The k-dimensional Weisfeiler–Leman algorithm is a powerful tool in graph isomorphism testing. For an input G, the determines canonical coloring of s-tuples vertices G for each s between 1 and k. We say that numerical parameter k-WL-invariant if it determined by tuple color. As application Dvo?ák’s result on k-WL-invariance homomorphism counts case k=2, we spot some non-obvious regularity properties strongly regular graphs related families. example, graph, then number 7-paths x y depends only whether or not are adjacent (this true also shorter paths but no longer 8-paths). Likewise, cycles length 7 passing through vertex same every (where optimal).