An infinite family of intransitive directed strongly regular graphs with rank 6 Weisfeiler--Leman closure

Abstract

We construct an infinite family of directed strongly regular graphs (DSRGs) with intransitive full automorphism groups whose Weisfeiler--Leman closure are association schemes of rank 6. This is the smallest possible rank for an association scheme admitting a proper DSRG merging. We also exhibit rigid sporadic DSRGs with the same closure property, showing that combinatorial regularity, group-theoretic symmetry, and Weisfeiler--Leman regularity capture fundamentally different aspects of graph structure.