Salem numbers defined by Coxeter transformation (Q1044592)

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 `\(a_{2n}\neq 0\)'