## Coxeter polynomials of Salem trees

### Volume 141 / 2015

Colloquium Mathematicum 141 (2015), 209-226
MSC: Primary 20F55.
DOI: 10.4064/cm141-2-6

#### Abstract

We compute the Coxeter polynomial of a family of Salem trees, and also the limit of the spectral radii of their Coxeter transformations as the number of their vertices tends to infinity. We also prove that if $z$ is a root of multiplicities $m_1,\ldots ,m_k$ for the Coxeter po\discretionary {-}{}{}ly\discretionary {-}{}{}no\discretionary {-}{}{}mials of the trees $\mathcal {T}_1,\ldots ,\mathcal {T}_k$ respectively, then $z$ is a root for the Coxeter po\discretionary {-}{}{}ly\discretionary {-}{}{}no\discretionary {-}{}{}mial of their join, of multiplicity at least $\min\{m-m_1,\ldots ,m-m_k\}$ where $m=m_1+\cdots +m_k$.