An entropy bridge from weighted to spectral Turán theorems
Abstract
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≥ 3, there exists a constant λ0=λ0(F) such that if G is an F-free graph with λ(G)≥ λ0, then for every ℓ≥ 1 with (r,ℓ )≠ (2,2), λℓ (G) ≤ (1−1/r)wℓ(G), with equality if and only if G is a regular complete r-partite graph; in the case r=2 with ℓ even, equality holds for every complete bipartite graph. The pair (r,ℓ )=(2,2) must be excluded, since the bound fails for several forbidden graphs, e.g., C2t+1 with t≥ 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 Kr+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 p is a probability vector on V(G) with lVert prVert∞ sufficiently small, then 2∑uv∈ E(G)pupv ≤ 1−1/r + 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.
Related
- Vertex-colored Turán theorems with applications in extremal hypergraph problems
- Weighted Turán densities for hypergraphs
- A counterexample to the Fang--Lin conjecture: Edge and spectral extremality diverge near a Turán graph
- The signless Laplacian spectral Turán problems for disjoint triangles
- Eigenvalue comparison theorems for the Witten-Laplacian and the weighted p-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
BibTeX
@misc{li2026entropy,
title = {{An entropy bridge from weighted to spectral Turán theorems}},
author = {Yongtao Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24388},
eprint = {2608.24388},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24388},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An entropy bridge from weighted to spectral Turán theorems
AU - Yongtao Li
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2608.24388
UR - https://arxiv.org/abs/2608.24388
AB - 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≥ 3, there exists a constant λ_0=λ_0(F) such that if G is an F-free graph with λ(G)≥ λ_0, then for every ℓ≥ 1 with (r,ℓ )≠ (2,2), λ^ℓ (G) ≤ (1−1/r)w_ℓ(G), with equality if and only if G is a regular complete r-partite graph; in the case r=2 with ℓ even, equality holds for every complete bipartite graph. The pair (r,ℓ )=(2,2) must be excluded, since the bound fails for several forbidden graphs, e.g., C_2t+1 with t≥ 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 p is a probability vector on V(G) with lVert prVert_∞ sufficiently small, then 2∑_uv∈ E(G)p_up_v ≤ 1−1/r + 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.
ER -