Salem numbers defined by Coxeter transformation
type: Item
instance of: scholarly article
title: Salem numbers defined by Coxeter transformation
zbMATH Open document ID: 1281.11093
author: Piroska Lakatos
published in: Linear Algebra and its Applications
publication date: 2009-12-18T00:00:00Z
full work available at URL: 1252082.pdf
review text: A generalised star is a simple connected graph that is not a tree, not a single cycle and has exactly one vertex of degree at least \(3\). Such a graph is made into an oriented \(n\)-vertex graph \(Q\) by giving each edge a unique orientation, but not allowing oriented cycles. Its adjacency matrix \(B\) is then an \(n\times n\) matrix that has a \(1\) if there is an edge from \(i\) to \(j\), and \(0\) otherwise. The Coxeter transformation \(c_Q\) of \(Q\) is then the matrix \(\Phi_Q= -(E-B)^{-1}(E-B)^T\), where \(E\) is the \(n\times n\) identity matrix. This is an integer matrix, since the `no oriented cycles' condition implies that the vertices of \(Q\) can be labelled so that \(B\) is strictly upper-triangular. The characteristic polynomial of \(Q\) is its Coxeter polynomial. The main result of this paper is that this Coxeter polynomial is the product of a Salem number and certain cyclotomic polynomials. Thus the spectral radii of such polynomials are Salem numbers. The proof is a detailed analysis of the roots of the Chebyshev transform (trace polynomial) of the Coxeter polynomial. Remark: At the bottom of page 145 `\(a_{2n}=0\)' should read `\(a_{2n}\neq 0\)'
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 5650026
zbMATH Keywords: Coxeter polynomial
zbMATH Keywords: Salem number
zbMATH Keywords: semi-reciprocal polynomials
zbMATH Keywords: Chebyshev transform
MaRDI profile type: Publication
OpenAlex ID: W2008591119
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DOI: 10.1016/J.LAA.2009.07.037
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description: scientific article; zbMATH DE number 5650026