Trees with Prescribed Maximum Degree and Spectral Radius

Abstract

It is well known that the spectral radius $\rho(T)$ of a tree $T$ with at least $3$ vertices satisfies $\frac 14\rho(T)^2+1<\Delta(T)\le \rho(T)^2$, where $\Delta(T)$ 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 $\alpha\in \mathbb{P}$ and every integer $r$ satisfying $\frac 14\alpha^2+1<r\le \alpha^2$, there exists a tree $T$ such that $\Delta(T)=r$ and $\rho(T)=\alpha$. For any positive integer $r$ and positive real number $\alpha$, define ${\mathscr W}_r(\alpha)$ recursively as follows. Initially, $\alpha\in {\mathscr W}_r(\alpha)$. Next, for any multiset $\left \{q_i: 1\le i\le s \right \}$ of positive elements of ${\mathscr W}_r(\alpha)$ with $q:=\alpha-\sum\limits_{i=1}^sq_i^{-1}\ge 0$, if either $s<r$ and $q\ge 0$, or $s=r$ and $q=0$, then $q\in {\mathscr W}_r(\alpha)$. We prove that $0\in {\mathscr W}_r(\alpha)$ if and only if there exists a tree $T$ with $\Delta(T)\le r$ and $\rho(T)=\alpha$. Consequently, $\mathbb{P}$ is exactly the set of positive numbers $\alpha$ such that $0\in {\mathscr W}_{\lfloor\alpha^2\rfloor}(\alpha)$. As an application, we show that for integers $k,r\ge 2$, there exists a tree $T$ with $\Delta(T)=r$ and $\rho(T)=\sqrt k$ if and only if $\frac 14 k+1<r\le k$.