Classification of partially metric Q-polynomial association schemes with $m_1 = 4$

TL;DR

This paper classifies a specific type of algebraic combinatorial structures called partially metric Q-polynomial association schemes with a particular parameter value, enhancing understanding of their structure and properties.

Contribution

It provides a complete classification of partially metric Q-polynomial association schemes with m_1=4, a previously unexplored parameter setting.

Findings

Classification of schemes with m_1=4 completed

Characterization of the relation between scheme graphs and distance graphs

Extension of known theory to new parameter regime

Abstract

We classify the Q-polynomial association schemes with $m_{1} = 4$ which are partially metric with respect to the nearest neighbourhood relation. An association scheme is partially metric with respect to a relation $R_1$ if the scheme graph of $R_2$ is exactly the distance-2 graph of the scheme graph of $R_1$ under a certain ordering of the relations.