Almost amorphic association schemes

TL;DR

This paper investigates the properties of amorphic association schemes, constructs non-amorphic examples with specific relations, and explores the coexistence of different strongly regular graph types within schemes.

Contribution

It constructs non-amorphic association schemes with two non-Latin square type relations for any number of classes and examines conditions for coexistence of strongly regular graph types.

Findings

Constructed non-amorphic schemes with two special relations for all classes d ≥ 4.

Showed that if a relation is a lattice graph, others must be Latin square type.

Raised open questions about coexistence of different strongly regular graph types.

Abstract

An association scheme is called amorphic if every possible fusion of relations gives rise to another association scheme. In earlier work, we showed that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it must be amorphic. We now construct non-amorphic $d$-class association schemes in which precisely two relations are not strongly regular of Latin square type or strongly regular of negative Latin square type, for any $d \geq 4$. We also raise the question whether different types of strongly regular graphs can coexist in an association scheme. Among some other results, we show that if one of the relations is a lattice graph, then any other strongly regular relation in the scheme must be of Latin square type.