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New graphs of finite mutation type. (English) Zbl 1180.05052

Summary: To a directed graph without loops or 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 S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No.2, 497–529 (2002; Zbl 1021.16017); Invent. Math. 154, No.1, 63–121 (2003; Zbl 1054.17024) ]. The mutation class of a graph \(\Gamma\) is the set of all isomorphism classes of graphs that can be obtained from \(\Gamma\) by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. S. Fomin, M. Shapiro, and D. Thurston [Acta Math. 201, No.1, 83–146 (2008; Zbl 1263.13023)] 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.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C76 Graph operations (line graphs, products, etc.)
05E99 Algebraic combinatorics
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