A unified spectral bound for color-critical graphs via a weighted Tur\'{a}n theorem

Abstract

In this paper, we establish the entropy-Perron bridge, and then use it to prove that for any color-critical graph $F$ with chromatic number $\chi(F)=r+1\ge 4$, there exists a constant $\lambda_0=\lambda_0(F)$ such that if $G$ is an $F$-free graph with $\lambda (G)\ge \lambda_0$, then for every $\ell\ge 1$, \[ \lambda^\ell (G) \le \Big(1-\frac1r\Big)w_\ell(G), \] with equality if and only if $G$ is a regular complete $r$-partite graph, possibly together with isolated vertices when $\ell \ge 2$. For odd $\ell$, the hypothesis can be relaxed to $\chi(F)\ge 3$, while at $\ell =2$, it cannot, since the bound fails for several forbidden graphs, e.g., $C_{2t+1}$ with $t\ge 2$. Moreover, walks may also be replaced by the homomorphism counts of unbalanced trees. As further applications, we extend a spectral supersaturation result of Bollob\'{a}s and Nikiforov [J. Combin. Theory Ser. B (2007)], and also extend the entropic Tur\'{a}n 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 Tur\'an 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 $\chi (F)-1$. The proof of this weighted result combines a blow-up argument, the graph removal lemma, the Erd\H{o}s-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.