  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.
