2026/04/30 by Fengming Dong, Ruixue Zhang
Mathematics · #math.CO #msc:05C05 #msc:05C20 #msc:05C50
15 pages and 2 figures
arxiv created 2026/08/05 · arxiv updated 2026/08/06
For any positive integer r and real number α>1, let \mathscr Lr(α) denote the set of positive real numbers defined recursively: α-1∈ \mathscr Lr(α) and, for any multi-set \q1,q2,…,qs\ of \mathscr Lr(α), where s<r,β:=α-1-s-∑i=1sqi-1 belongs to \mathscr Lr(α) as long as β>0. We first show that there exists a tree T with maximum degree Δ(T)≤ r and the Laplacian spectral radius μ(T)=α if and only if (α-1)-1∈ \mathscr Lr(α). It follows that the set of Laplacian spectral radii of non-trivial trees is exactly the set of real numbers α≥ 2 such that (α-1)-1∈ \mathscr Lr(α), where r=\lfloorα\rfloor-1. Applying this result, we show that for any integer k≥ 2, there exists a tree T with μ(T)=k2 and Δ(T)=r if and only if (k-1)2+2≤ r≤ k2-1.