A new characterization of the set of Laplacian spectral radii of trees

Abstract

For any positive integer $r$ and real number $\alpha>1$, let ${\mathscr L}_r(\alpha)$ denote the set of positive real numbers defined recursively: $\alpha-1\in {\mathscr L}_r(\alpha)$, and for any multi-subset $\{q_1,q_2,\dots,q_s\}$ of ${\mathscr L}_r(\alpha)$, where $0<s<r$, $\beta:=\alpha-1-s-\sum_{i=1}^sq_i^{-1}$ belongs to ${\mathscr L}_r(\alpha)$ as long as $\beta>0$. We show that $(\alpha-1)^{-1}\in {\mathscr L}_r(\alpha)$ if and only if there exists a tree $T$ with its maximum degree $\Delta(T)\le r$ and Laplacian spectral radius $\mu(T)=\alpha>1$. It follows that the set of Laplacian spectral radii of non-trivial trees is exactly the set of real numbers $\alpha\ge 2$ such that $(\alpha-1)^{-1}\in {\mathscr L}_r(\alpha)$ for $r=\lfloor\alpha\rfloor-1$. Applying this conclusion, we then show that for any integer $n$, there exists a tree $T$ with $\Delta(T)<n$ and $\mu(T)=n+1$ if and only if $n\ge 4$.