On Coxeter polynomials of wild stars

Lakatos, Piroska (1999) On Coxeter polynomials of wild stars. LINEAR ALGEBRA AND ITS APPLICATIONS, 293. pp. 159-170. ISSN 0024-3795

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Abstract

The spectral radius of a Coxeter transformation which plays an important role in the representation theory of hereditary algebras [see V. Dlab, C.M. Ringel, Eigenvalues of Coxeter transformations and the Gelfand±Kirillov dimension of the preprojective algebras, Proc. AMS 83 (1990) 228±232] is its important invariant. This paper provides both upper and lower bounds for the spectral radii of the Coxeter transformations of wild stars (i.e. trees that have a single branching point and are neither of Dynkin nor of Euclidean type). In addition, the paper determines limit of the spectral radii of particular in®nite sequences of wild stars and shows different classes of graphs with the same limit. The basic idea is to reduce the study of spectral radii of trees to the spectral radii of particular valued graphs with inde®nite type of associated generalized Cartan matrix. Ó 1999 Elsevier Science Inc. All rights reserved.

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