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Motivated by string diagrammatic approach to undirected tracial quantumgraphs by Musto, Reutter, Verdon (2018), in the former part of this paper wediagrammatically formulate directed nontracial quantum graphs by Brannan,Chirvasitu, Eifler, Harris, Paulsen, Su, Wasilewski (2019). In the latter part,we supply a concrete classification of undirected reflexive quantum graphs on$M_2$ and their quantum automorphism groups in both tracial and nontracialsettings. We also obtain quantum isomorphisms between tracial quantum graphs on$M_2$ and certain classical graphs, which reproves the monoidal equivalencesbetween $SO(3)$ and $S_4^+$, and $O(2)$ and $H_2^+$.

Classification of Quantum Graphs on $M_2$ and their Quantum Automorphism Groups