Local WL Invariance and Hidden Shades of Regularity

Frank Fuhlbrück; Johannes Köbler; Oleg Verbitsky

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arXiv · 2002.04590

Local WL Invariance and Hidden Shades of Regularity

Frank Fuhlbrück·Johannes Köbler·Oleg Verbitsky

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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 Dvo\v{r}\'ak's result on $k$-WL-invariance of homomorphism counts, 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 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). Or, 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).

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Frank Fuhlbrück, Johannes Köbler, Oleg Verbitsky. 2020-02-11. Local WL Invariance and Hidden Shades of Regularity. https://arxiv.org/abs/2002.04590

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