带规模约束的图的谱半径：解决 Guiduli 猜想
Spectral Radius of Graphs with Size Constraints: Resolving a Conjecture of Guiduli
AI总结：
本文解决了 Guiduli 关于满足遗传有界性质 $P_{t,r}$ 的图谱半径猜想，通过引入新颖势函数并结合边平移操作证明了谱半径上界，且完整刻画了达到该界限的极值图结构。
摘要：
我们解决了 Guiduli (1996) 提出的关于满足遗传有界性质 $P_{t,r}$ 的图谱半径的问题，该性质要求每个满足 $|V(H)| \geq t$ 的子图 $H$ 满足 $|E(H)| \leq t|V(H)| + r$。对于满足 $P_{t,r}$ 的 $n$ 顶点图 $G$，其中 $t > 0$ 且 $r \geq -\binom{\lfloor t+1 \rfloor}{2}$，我们证明谱半径 $ρ(G)$ 的上界为 $ρ(G) \leq c(s,t) + \sqrt{\lfloor t \rfloor n}$，其中 $s = \binom{\lfloor t \rfloor + 1}{2} + r$，从而对 Guiduli 的猜想给出了肯定回答。此外，我们对达到此界限的极值图进行了完整刻画。这些图被构造为联图 $K_{\lfloor t \rfloor} \nabla F$，其中 $F$ 为 $K_3 \cup (n - \lfloor t \rfloor - 3)K_1$ 或仅由星型结构组成的森林。我们细致地刻画了此类森林的具体结构。我们分析的核心在于引入了一个新颖的势函数 $η(F) = e(F) + (\lfloor t \rfloor - t)|V(F)|$，用于量化子图的结构“正性”。通过将边平移操作与谱半径最大化原理相结合，我们确立了 $η^+(G)$（即 $G$ 的累积正性）的严格界限。我们的结果有助于理解边密度约束下的谱极值问题，并为分析类似遗传性质提供了框架。
英文摘要：
We resolve a problem posed by Guiduli (1996) on the spectral radius of graphs satisfying the Hereditarily Bounded Property $P_{t,r}$, which requires that every subgraph $H$ with $|V(H)| \geq t$ satisfies $|E(H)| \leq t|V(H)| + r$. For an $n$-vertex graph $G$ satisfying $P_{t,r}$, where $t > 0$ and $r \geq -\binom{\lfloor t+1 \rfloor}{2}$, we prove that the spectral radius $ρ(G)$ is bounded above by $ρ(G) \leq c(s,t) + \sqrt{\lfloor t \rfloor n}$, where $s = \binom{\lfloor t \rfloor + 1}{2} + r$, thus affirmatively answering Guiduli's conjecture. Furthermore, we present a complete characterization of the extremal graphs that achieve this bound. These graphs are constructed as the join graph $K_{\lfloor t \rfloor} \nabla F$, where $F$ is either $K_3 \cup (n - \lfloor t \rfloor - 3)K_1$ or a forest consisting solely of star structures. The specific structure of such forests is meticulously characterized. Central to our analysis is the introduction of a novel potential function $η(F) = e(F) + (\lfloor t \rfloor - t)|V(F)|$, which quantifies the structural "positivity" of subgraphs. By combining edge-shifting operations with spectral radius maximization principles, we establish sharp bounds on $η^+(G)$, the cumulative positivity of $G$. Our results contribute to the understanding of spectral extremal problems under edge-density constraints and provide a framework for analyzing similar hereditary properties.
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