A Recursive Characterization of Laplacian Spectral Radii of Trees with Bounded Maximum Degree

Abstract

For a positive integer $r$ and a real number $\alpha \ge 2$, let ${\mathcal{L}}_r(\alpha )$ be the set of positive real numbers containing $\alpha -1$ and closed under the following operation: if $q_1,\dots ,q_s\in {\mathcal{L}}_r(\alpha )$, where $1\le s\le r-1$, and $q=\alpha -1-s-\sum _{i=1}^s{q}_i^{-1}>0$, then $q\in {\mathcal{L}}_r(\alpha )$. We prove that there exists a tree $T$ with $\Delta (T)\le r$ and Laplacian spectral radius $\mu (T)=\alpha$ if and only if $(\alpha -1{)}^{-1}\in {\mathcal{L}}_r(\alpha )$. Consequently, the Laplacian spectral radii of nontrivial trees are precisely the real numbers $\alpha \ge 2$ satisfying $(\alpha -1{)}^{-1}\in {\mathcal{L}}_{\lfloor \alpha \rfloor -1}(\alpha )$. As an application, we prove that, for integers $k\ge 2$ and $r\ge 2$, a tree $T$ with $\mu (T)=k^2$ and $\Delta (T)=r$ exists if and only if $(k-1{)}^2+2\le r\le k^2-1$.

18 pages and 3 figures