The Weisfeiler-Leman algorithm and the diameter of Schreier graphs

TL;DR

This paper explores the relationship between the Weisfeiler-Leman algorithm's iteration count and the diameter of Schreier graphs, providing bounds and exact formulas for specific cases including Cayley graphs.

Contribution

It establishes bounds linking Weisfeiler-Leman iterations to graph diameter and derives exact formulas for Cayley graphs, advancing understanding of graph isomorphism testing.

Findings

Upper bound for Weisfeiler-Leman iterations on Schreier graphs

Lower bound for specific Schreier graphs with SL_n(F_q)

Exact expression for Cayley graphs

Abstract

We prove that the number of iterations taken by the Weisfeiler-Leman algorithm for configurations coming from Schreier graphs is closely linked to the diameter of the graphs themselves: an upper bound is found for general Schreier graphs, and a lower bound holds for particular cases, such as for Schreier graphs with $G=\mbox{SL}_{n}({\mathbb F}_{q})$ ($q>2$) acting on $k$-tuples of vectors in ${\mathbb F}_{q}^{n}$; moreover, an exact expression is found in the case of Cayley graphs.