# Local WL Invariance and Hidden Shades of Regularity

The k-dimensional Weisfeiler-Leman algorithm (k-WL) is a powerful tool for testing isomorphism of two given graphs. We aim at investigating the ability of k-WL to capture properties of vertices (or small sets of vertices) in a single input graph G. In general, k-WL computes a canonical coloring of k-tuples of vertices of G, which determines a canonical coloring of s-tuples for each s between 1 and k. We say that a property (or a numerical parameter) of s-tuples is k-invariant if it is determined by the tuple color. Our main result establishes k-invariance of the parameters counting the number of extensions of an s-tuple of vertices to a given subgraph pattern F. We state a sufficient condition for k-invariance in terms of the treewidth of F and its homomorphic images, using suitable variants of these concepts for graphs with s designated roots. As an application, we observe some non-obvious regularity properties of strongly regular graphs: For example, if G is strongly regular, then the number of paths of length 6 between vertices x and y in G depends only on whether or not x and y are adjacent (and the length 6 is here optimal). Despite the fact that k-WL indistinguishability of vertex tuples implies high degree of regularity, we prove, on the negative side, that no fixed dimension k suffices for k-WL to recognize global symmetry of a graph. Specifically, for every k, there is a graph G whose vertex set is colored by k-WL uniformly while G is not vertex-transitive.

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