Trees with Prescribed Maximum Degree and Spectral Radius

Abstract

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 $α\in \mathbb{P}$ 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 any positive integer $r$ and positive real number $α$, define ${\mathscr W}_r(α)$ recursively as follows. Initially, $α\in {\mathscr W}_r(α)$. Next, for any multiset $\left \{q_i: 1\le i\le s \right \}$ of positive elements of ${\mathscr W}_r(α)$ with $q:=α-\sum\limits_{i=1}^sq_i^{-1}\ge 0$, if either $s<r$ and $q\ge 0$, or $s=r$ and $q=0$, then $q\in {\mathscr W}_r(α)$. We prove that $0\in {\mathscr W}_r(α)$ if and only if there exists a tree $T$ with $Δ(T)\le r$ and $ρ(T)=α$. Consequently, $\mathbb{P}$ is exactly the set of positive numbers $α$ such that $0\in {\mathscr W}_{\lfloorα^2\rfloor}(α)$. 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 $\frac 14 k+1<r\le k$.