无kK₃子图的图的严格固定大小谱上界
A sharp fixed-size spectral bound for $kK_3$-free graphs
浏览论文内容
中文总结 AI 辅助
本文针对固定整数k≥2，建立无kK₃子图的m边图的严格邻接谱上界，证明等号成立的条件与图结构，k=2的情况此前已知，本文将结论推广至所有k≥3，并说明需二阶分析的原因。
中文摘要 AI 辅助
对于固定整数k≥2，我们建立了足够多边数为m的无kK₃子图的邻接谱严格上界，证明λ(G)≤(k-1)+√(m-k(k-1))。当且仅当(2k-1)整除m时等号成立，且除去孤立顶点外，G是K_{2k-1}与大小为m/(2k-1)-(k-1)的独立集的联图。k=2的情况此前已有结论，本文论证适用于所有固定k≥3。证明需用到一阶谱稳定性之外的信息：我们在最大Perron顶点处推导精确非负缺陷恒等式，用其将整个外层约束为常数，再将剩余图归约为具有有限个独立孪生类的有界核心；随后利用Perron向量集中恒等式与Erdős–Gallai匹配定理得到唯一极值核心。近极值族仅比目标低Θ(m^{-1/2})，这解释了为何需要精确二阶分析。
英文摘要
For a fixed integer $k\ge2$, we establish a sharp adjacency-spectral upper bound for sufficiently large $m$-edge $kK_3$-free graphs. We prove \[ λ(G)\le (k-1)+\sqrt{m-k(k-1)}. \] Moreover, equality holds precisely when $(2k-1)\mid m$ and, up to isolated vertices, $G$ is the join of $K_{2k-1}$ with an independent set of $m/(2k-1)-(k-1)$ vertices. The case $k=2$ was previously known; our argument establishes every fixed $k\ge3$. The proof requires information beyond first-order spectral stability. We derive an exact nonnegative defect identity at a maximum Perron vertex, use it to bound the entire outer layer by a constant, and reduce the remaining graph to a bounded core with finitely many independent twin classes. A Perron-vector concentration identity and the Erdős--Gallai matching theorem then force the unique extremal core. A nearly extremal family lies only $Θ(m^{-1/2})$ below the target, showing why an exact second-order analysis is necessary.
