On the WL-dimension of circulant graphs of prime power order

TL;DR

This paper proves that the Weisfeiler-Leman dimension of any circulant graph with prime power order is at most 3, establishing a tight bound using coherent configurations and Cayley schemes.

Contribution

It establishes a tight upper bound of 3 for the WL-dimension of circulant graphs of prime power order, a result not previously known.

Findings

WL-dimension of such graphs is at most 3

The bound is proven to be tight

Uses theories of coherent configurations and Cayley schemes

Abstract

The WL-dimension of a graph X is the smallest positive integer m such that the m-dimensional Weisfeiler-Leman algorithm correctly tests the isomorphism between X and any other graph. It is proved that the WL-dimension of any circulant graph of prime power order is at most 3, and this bound cannot be reduced. The proof is based on using theories of coherent configurations and Cayley schemes over a cyclic group.