发表机构
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立熵-佩龙桥梁，结合加权Turán定理框架，证明了色临界图的统一谱界，推广了相关谱超饱和与熵Turán定理结果。
AI 中文摘要
本文建立了熵-佩龙（entropy-Perron）桥梁，并利用它证明：对于任意色数χ(F)=r+1≥4的色临界图F，存在常数λ₀=λ₀(F)，使得若G是不含F的图且其谱半径λ(G)≥λ₀，则对每个ℓ≥1，有λ^ℓ(G)≤(1−1/r)w_ℓ(G)，等号成立当且仅当G是正则完全r部图，当ℓ≥2时可附带孤立顶点。对于奇数ℓ，假设可放宽至χ(F)≥3；而当ℓ=2时无法放宽，因为该界对若干禁图不成立，例如t≥2时的C_{2t+1}。此外，游走（walks）也可替换为不平衡树的同态计数。作为进一步应用，我们推广了Bollobás和Nikiforov在《组合理论杂志B辑》（2007年）中的谱超饱和结果，还将Chao和Yu在《伦敦数学会杂志》（2026年）中针对不含K_{r+1}图的熵Turán定理，推广至不含色临界图F的图。我们通过具有独立意义的加权Turán定理构建了一个框架：若G不含F，且p是V(G)上的概率向量，其无穷范数‖p‖_∞足够小，则2∑_{uv∈E(G)}p_up_v ≤1−1/r + o(1)，且仅当F是色临界图时可去除误差项o(1)。这是一类Motzkin-Straus型不等式，其中G的团数被替换为χ(F)−1。该加权结果的证明结合了膨胀论证、图删除引理、Erdős-Simonovits稳定性定理、概率抽样论证以及完全r部图附近的精确估计。连接谱不等式与加权不等式的桥梁基于与佩龙向量关联的马尔可夫链的熵方法。
英文摘要
We establish an entropy bridge, and then use it to prove that for any color-critical graph $F$ with chromatic number $χ(F)=r+1\ge 3$, there exists a constant $λ_0=λ_0(F)$ such that if $G$ is an $F$-free graph with $λ(G)\ge λ_0$, then for every $\ell\ge 1$ with $(r,\ell )\neq (2,2)$, \[ λ^\ell (G) \le \Big(1-\frac1r\Big)w_\ell(G), \] with equality if and only if $G$ is a regular complete $r$-partite graph; in the case $r=2$ with $\ell$ even, equality holds for every complete bipartite graph. The pair $(r,\ell )=(2,2)$ must be excluded, since the bound fails for several forbidden graphs, e.g., $C_{2t+1}$ with $t\ge 2$. Moreover, walks counts may also be replaced by the homomorphism counts of unbalanced trees. As further applications, we extend a spectral supersaturation result of Bollobás and Nikiforov [J. Combin. Theory Ser. B (2007)], and we also extend the entropic Turán theorem of Chao and Yu [J. London Math. Soc. (2026)] from $K_{r+1}$-free graphs to $F$-free graphs with $F$ color-critical. We provide a framework by passing through weighted Turá theorems of independent interest. If $G$ is $F$-free and $\mathbf{p}$ is a probability vector on $V(G)$ with $\lVert \mathbf{p}\rVert_\infty$ sufficiently small, then \[ 2\sum_{uv\in E(G)}p_up_v \le 1-\frac1r + o(1), \] and the error term $o(1)$ can be removed if and only if $F$ is color-critical. This is a Motzkin-Straus-type inequality in which the clique number of $G$ is replaced by $χ(F)-1$. The proof of this weighted result combines a blow-up argument, the graph removal lemma, the Erdős-Simonovits stability theorem, a probabilistic sampling argument, and an exact estimate near a complete $r$-partite graph. The bridge linking spectral inequalities to weighted inequalities is based on the entropy method for the Markov chain attached to the Perron vector.
Comments41 pages, 1 table, and 1 figure. We changed the title and solved the missing case in 2nd version
