Derksen, Harm; Owen, Theodore New graphs of finite mutation type. (English) Zbl 1180.05052 Electron. J. Comb. 15, No. 1, Research Paper R139, 15 p. (2008). 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. Cited in 2 ReviewsCited in 13 Documents MSC: 05C20 Directed graphs (digraphs), tournaments 05C76 Graph operations (line graphs, products, etc.) 05E99 Algebraic combinatorics Keywords:directed graph; skew symmetric matrix; mutations of skew symmetric matrices; mutations of skew symmetrizable matrices; mutation class; mutation finite graph; graphs of geometric type; exceptional mutation classes; finite mutation classes Citations:Zbl 1021.16017; Zbl 1054.17024; Zbl 1263.13023 PDFBibTeX XMLCite \textit{H. Derksen} and \textit{T. Owen}, Electron. J. Comb. 15, No. 1, Research Paper R139, 15 p. (2008; Zbl 1180.05052) Full Text: arXiv EuDML EMIS