Four-vertex quivers supporting twisted graded Calabi-Yau algebras

We study quivers supporting twisted graded Calabi-Yau algebras, building on work of Rogalski and the first author. Specifically, we classify quivers on four vertices in which the Nakayama automorphism acts on the degree zero part by either a four-cycle, a three-cycle, or two two-cycles. In order to realize algebras associated to some of these quivers, we show that graded twists of a twisted graded Calabi-Yau algebra is another algebra of the same type.