An Upper Bound for the Height of a Tree with a Given Eigenvalue

In this paper we prove that every totally real algebraic integer \(\lambda \) $\lambda$ of degree \(d \ge 2\) $d\ge 2$ occurs as an eigenvalue of some tree of height at most \(d(d+1)/2+3\) $d(d+1)/2+3$ . In order to prove this, for a given algebraic number \(\alpha \ne 0\) $\alpha \ne 0$ , we investigate an additive semigroup that contains zero and is closed under the map \(x \mapsto \alpha /(1-x)\) $x\mapsto \alpha /(1-x)$ for \(x \ne 1\) $x\ne 1$ . The problem of finding the smallest such semigroup seems to be of independent interest.