New Graphs of Finite Mutation Type

To a directed graph without loops and 2-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph G is the set of all isomorphism classes of graphs that can be obtained from G by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. Fomin, Shapiro and Thurston constructed mutation-finite graphs from triangulations of oriented bordered surfaces with marked points. We will call such graphs "of geometric type". Besides graphs with 2 vertices, and graphs of geometric type, there are only 9 other "exceptional" mutation classes that are known to be finite. In this paper we introduce 2 new exceptional finite mutation classes.