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arXiv · 2604.09972

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

Fengming Dong·Ruixue Zhang
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Abstract

For a positive integer $r$ and a real number $\alpha\ge 2$, let $\mathscr L_r(\alpha)$ be the set of positive real numbers containing $\alpha-1$ and closed under the following operation: if $q_1,\ldots,q_s\in\mathscr L_r(\alpha)$, where $1\leq s\leq r-1$, and $q=\alpha-1-s-\sum\limits_{i=1}^{s}q_i^{-1}>0$, then $q\in\mathscr L_r(\alpha)$. We prove that there exists a tree $T$ with $\Delta(T)\leq r$ and Laplacian spectral radius $\mu(T)=\alpha$ if and only if $(\alpha-1)^{-1}\in\mathscr L_r(\alpha)$. Consequently, the Laplacian spectral radii of nontrivial trees are precisely the real numbers $\alpha\geq2$ satisfying $(\alpha-1)^{-1}\in\mathscr L_{\lfloor\alpha\rfloor-1}(\alpha)$. As an application, we prove that, for integers $k\geq2$ and $r\geq2$, a tree $T$ with $\mu(T)=k^2$ and $\Delta(T)=r$ exists if and only if $(k-1)^2+2\leq r\leq k^2-1$.

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Fengming Dong, Ruixue Zhang. 2026-04-11. A Recursive Characterization of Laplacian Spectral Radii of Trees with Bounded Maximum Degree. https://arxiv.org/abs/2604.09972

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