Complete classification of directed quantum graphs on $$M_2$$

Abstract In 2022, Gromada and Matsuda classified undirected quantum graphs on the matrix algebra $$M_2$$ M 2 (Gromada in Lett Math Phys 112:122, 2022; Matsuda in J Math Phys. 63:092201, 2022). Later, Wasilweski provided a solid theory of directed quantum graphs (Wasilewski in 29:1281–1317, 2024) which was formerly only established for undirected quantum graphs. Using this framework we extend the results of Matsuda and Gromada, and present a complete classification of directed quantum graphs on $$M_2$$ M 2 . Most prominently, we observe that there is a far bigger range of directed quantum graphs than of undirected quantum graphs on $$M_2$$ M 2 . Moreover, for quantum graphs on a nontracial quantum set $$(M_2, \psi )$$ ( M 2 , ψ ) we illustrate the difference between GNS- and KMS-undirected quantum graphs.