Classification of Partially Metric Q-Polynomial Association Schemes with $$m_{ 1} = 4$$

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.