  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.
