Local WL invariance and hidden shades of regularity

Abstract

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