arXiv2026
It is well known that the spectral radius $ρ(T)$ of a tree $T$ with at least $3$ vertices satisfies $\frac 14ρ(T)^2+1<Δ(T)\le ρ(T)^2$, where $Δ(T)$ is the maximum degree of $T$. Let $\mathbb{P}$ denote the set of spectral radii of all non-trivial trees. We ask whether, for every $α\in \mathbb{P}\setminus \{1\}$ and every integer $r$ satisfying $\frac 14α^2+1<r\le α^2$, there exists a tree $T$ such that $Δ(T)=r$ and $ρ(T)=α$. For a positive integer $r$ and a positive real number $α$, let ${\mathscr W}_r(α)$ be the smallest subset of $[0,\infty)$ containing $α$ and closed under the following operation: if $q_1,\ldots,q_s$ are positive elements of ${\mathscr W}_r(α)$, not necessarily distinct, and $q=α-\sum_{i=1}^{s}q_i^{-1}$, then $q\in{\mathscr W}_r(α)$ whenever either $1\le s<r$ and $q\ge0$, or $s=r$ and $q=0$. We prove that $0\in {\mathscr W}_r(α)$ if and only if there exists a tree $T$ with $Δ(T)\le r$ and $ρ(T)=α$. Moreover, there exists a tree $T$ with $Δ(T)=r$ and $ρ(T)=α$ if and only if there are positive elements $q_1,\ldots,q_r\in\mathscr W_r(α)$, not necessarily distinct, satisfying $\sum_{i=1}^{r}q_i^{-1}=α$. Consequently, $\mathbb{P}=\{α\geq1:0\in {\mathscr W}_{\lfloorα^2\rfloor}(α)\}.$ As an application, we show that for integers $k,r\ge 2$, there exists a tree $T$ with $Δ(T)=r$ and $ρ(T)=\sqrt k$ if and only if $\frac 14 k+1<r\le k$.