2025/04/09 by Fengming Dong, Dong, Fengming, Ruixue Zhang +1 · 1 citation
Computer Science · Mathematics · #05C05 #05C20 #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2504.06617
openalex publication_date 2025/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well known that the spectral radius ρ(T) of a tree T with at least 3 vertices has the property that \frac 14ρ(T)2+1<Δ(T)≤ ρ(T)2, where Δ(T) is the maximum degree of T. Let ℙ denote the set of spectral radii of all non-trivial trees. In this article, we study the inverse problem that for any α∈ ℙ and integer r satisfying the condition \frac 14α2+10, 1≤ i≤ s\, if β:=α-∑i=1sqi-1≥ 0 and s≤ r- \lceil \fracββ+1\rceil, then β∈ \mathscr Wr(α). We first show that 0∈ \mathscr Wr(α) if and only if there exists a tree T with Δ(T)≤ r and ρ(T)=α. It follows directly that ℙ is exactly the set of positive numbers α such that 0∈ \mathscr W\lfloorα2\rfloor(α). Applying this conclusion, we prove that for any two positive integers r≥ 2 and k, there exists a tree T with Δ(T)=r and ρ(T)=√ k if and only if \frac 14 k+1