Dimension two twisted graded Calabi--Yau algebras on two-vertex quivers

We classify, up to isomorphism, twisted graded Calabi–Yau algebras of dimension two on two-vertex quivers. By work of Reyes and Rogalski, such algebras may be presented as quotients of translation quivers by mesh relations. We also consider the isomorphism problem for certain families of twisted graded Calabi–Yau algebras on larger quivers.