  Motivated by string diagrammatic approach to undirected tracial quantum
graphs by Musto, Reutter, Verdon (2018), in the former part of this paper we
diagrammatically 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 nontracial
settings. We also obtain quantum isomorphisms between tracial quantum graphs on
$M_2$ and certain classical graphs, which reproves the monoidal equivalences
between $SO(3)$ and $S_4^+$, and $O(2)$ and $H_2^+$.
