具有预定最大度数和谱半径的树的存在性
组合数学
2025-09-15 v3
摘要
众所周知，至少有 3 个顶点的树 T T T ρ ( T ) \rho(T) ρ ( T ) 1 4 ρ ( T ) 2 + 1 < Δ ( T ) ≤ ρ ( T ) 2 \frac 14\rho(T)^2+1<\Delta(T)\le \rho(T)^2 4 1 ρ ( T ) 2 + 1 < Δ ( T ) ≤ ρ ( T ) 2 Δ ( T ) \Delta(T) Δ ( T ) T T T P \mathbb{P} P α ∈ P \alpha \in \mathbb{P} α ∈ P 1 4 α 2 + 1 < r ≤ α 2 \frac 14\alpha^2+1<r\le \alpha^2 4 1 α 2 + 1 < r ≤ α 2 r r r T T T Δ ( T ) = r \Delta(T)=r Δ ( T ) = r ρ ( T ) = α \rho(T)=\alpha ρ ( T ) = α r r r α \alpha α W r ( α ) {\mathscr W}_r(\alpha) W r ( α ) α ∈ W r ( α ) \alpha\in {\mathscr W}_r(\alpha) α ∈ W r ( α ) { q i ∈ W r ( α ) : q i > 0 , 1 ≤ i ≤ s } \{q_i\in {\mathscr W}_r(\alpha): q_i>0, 1\le i\le s\} { q i ∈ W r ( α ) : q i > 0 , 1 ≤ i ≤ s } β : = α − ∑ i = 1 s q i − 1 ≥ 0 \beta:=\alpha-\sum\limits_{i=1}^sq_i^{-1}\ge 0 β := α − i = 1 ∑ s q i − 1 ≥ 0 s ≤ r − ⌈ β β + 1 ⌉ s\le r-\left \lceil \frac{\beta}{\beta+1}\right \rceil s ≤ r − ⌈ β + 1 β ⌉ β ∈ W r ( α ) \beta \in {\mathscr W}_r(\alpha) β ∈ W r ( α ) 0 ∈ W r ( α ) 0\in {\mathscr W}_r(\alpha) 0 ∈ W r ( α ) T T T Δ ( T ) ≤ r \Delta(T)\le r Δ ( T ) ≤ r ρ ( T ) = α \rho(T)=\alpha ρ ( T ) = α P \mathbb{P} P 0 ∈ W ⌊ α 2 ⌋ ( α ) 0\in {\mathscr W}_{\lfloor\alpha^2\rfloor}(\alpha) 0 ∈ W ⌊ α 2 ⌋ ( α ) α \alpha α r ≥ 2 r\ge 2 r ≥ 2 k k k T T T Δ ( T ) = r \Delta(T)=r Δ ( T ) = r ρ ( T ) = k \rho(T)=\sqrt k ρ ( T ) = k 1 4 k + 1 < r ≤ k \frac 14 k+1<r\le k 4 1 k + 1 < r ≤ k
引用
@article{arxiv.2504.06617,
title = {Existence of trees with prescribed maximum degrees and spectral radii},
author = {Fengming Dong and Ruixue Zhang},
journal= {arXiv preprint arXiv:2504.06617},
year = {2025}
}
备注
17 pages and 2 figures