On the Coxeter polynomials of wild stars

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 infinite 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 indefinite type of associated generalized Cartan matrix.