On the Godsil -- Higman necessary condition for equitable partitions of association schemes

TL;DR

This paper examines Godsil's necessary condition for equitable partitions in association schemes and demonstrates that it is equivalent to the Lloyd theorem, clarifying its strength and limitations.

Contribution

The authors show that Godsil's necessary condition is not stronger than the Lloyd theorem, clarifying its role in the theory of equitable partitions.

Findings

Godsil's condition is equivalent to Lloyd's theorem.

The condition does not provide additional restrictions beyond Lloyd's theorem.

Clarifies the theoretical relationship between two key conditions in association schemes.

Abstract

In the monograph 'Association schemes', C. Godsil derived a necessary condition for equitable partitions of association schemes and noticed that it could be used to show that certain equitable partitions do not exist. In this short note, we show that, in fact, this condition is not stronger than the well-known Lloyd theorem.