Graph-symmetric Period 2 Quiver And It's Generalized Mutation

In 2002,Serger Fomin and Andrei Zelevinsky introduced the notion of cluster algebras,and proved the Laurent phenomenon of cluster algebras.Applying the Laurent phenomenon of cluster algebras,we can prove that all terms of Somos-4 sequence and Somos-5sequence are integer.The mutation-periodic quivers were defined by Allan P.Fordy and Robert J.Marsh in 2011.Allan P.Fordy and Robert J.Marsh classified period 1 quivers and gave families of quivers which have higher periodicity.The periodicity means that sequences given by recurrence relations arise from the associated cluster algebras.Every mutation-periodic quiver corresponds to a recurrence relation which has the same property with the recurrence relation of Somos-4 sequence and Somos-5 sequence.In this paper,we give a necessary and sufficient condition of period 2 property of graph-symmetric quivers,and generalize the conclusions of period 1 quivers and period 2 quivers to generalized cluster algebras.