无三角形图的高连通性谱Ore定理
A higher-connectivity spectral Ore theorem for triangle-free graphs
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中文总结 AI 辅助
该研究针对无三角形图，证明当n≥4k+2时，补图连通度不小于k的无三角形图中，B_{n,k}是使得ρA(G)取最大值的唯一极图，还求解了相关二部问题、边界值及k=1、2时的特殊情况。
中文摘要 AI 辅助
设B_{n,k}为从n顶点平衡完全二部图中删除大小为k的匹配得到的图。若G是n顶点无三角形图，且其补图的连通度κ(补G)≥k，我们证明当n≥4k+2时，ρA(G)≤ρA(B_{n,k})，等号成立当且仅当G≅B_{n,k}，且明确计算了ρA(B_{n,k})。我们还对所有n≥2k+1求解二部问题，确定边界值spex_κ(2k,K_3;k)=k-1，解决k=2时的完整问题，特别地，B_{n,2}从阶数6起为唯一极图；对于k=1，即补图连通时，B_{n,1}=K_{⌈n/2⌉,⌊n/2⌋}-e对所有n≥3为唯一极图。
英文摘要
Let $B_{n,k}$ be the graph obtained from the balanced complete bipartite graph on $n$ vertices by deleting a matching of size $k$. If $G$ is an $n$-vertex triangle-free graph with $κ(\comp G)\geq k$, we prove that $\rhoA(G)\leq\rhoA(B_{n,k})$ for $n\geq4k+2$, with equality precisely when $G\cong B_{n,k}$, and we compute $\rhoA(B_{n,k})$ explicitly. We also solve the bipartite problem for every $n\geq2k+1$, determine the boundary value $\operatorname{spex}_κ(2k,K_3;k)=k-1$, and settle the full problem for $k=2$. In particular, $B_{n,2}$ is uniquely extremal exactly from order $6$ onward. For $k=1$, equivalently when the complement is connected, $B_{n,1}=K_{\ceil{n/2},\floor{n/2}}-e$ is uniquely extremal for every $n\geq3$.
