A new spectral Turán theorem for weighted graphs and consequences

Confirming a conjecture of Elphick and Edwards and strengthening a spectral theorem of Wilf, Nikiforov proved that for any $K_{r+1}$-free graph $G$, $λ(G)^2 \leq 2 (1 - 1/r) m$, where $λ(G)$ is the spectral radius of $G$, and $m$ is the number of edges of $G$. This result was later improved in \cite{LiuN26}, where it was shown that for any graph $G$, $λ(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\mathrm{cl}(e) - 1}{\mathrm{cl}(e)}$, where $\mathrm{cl}(e)$ denotes the order of the largest clique containing the edge $e$. In this paper, we further extend this inequality to weighted graphs, proving that [ λ(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\mathrm{cl}(e) - 1}{\mathrm{cl}(e)} w(e)^2, ] and we characterize all extremal graphs attaining this bound. Our main theorem yields several new consequences, including two vertex-based and vertex-degree-based local versions of Turán’s theorem, as well as weighted generalizations of the Edwards–Elphick theorem and the Cvetković theorem, and two localized versions of Wilf’s theorems. One of these localized Wilf’s theorem confirms a conjecture that originates from Probability and Operator Algebras and was proposed by R. Tripathi independently of us. Moreover, our main result unifies and implies numerous earlier ones from spectral graph theory and extremal graph theory, including Stanley’s spectral inequality, Hong’s inequality, a localized Turán-type theorem, and a recent extremal theorem by Adak and Chandran. Notably, while Nikiforov’s earlier spectral inequality implied Stanley’s bound, it did not imply Hong’s inequality – a gap that is now bridged by our result. As a key tool, we establish the inequality $\sum_{e \in E(G)} \frac{2}{\mathrm{cl}(e)} \geq n-1$, which complements an upper bound $\sum_{e \in E(G)} \frac{2}{\mathrm{cl}(e)-1} \leq n^2 - 2m$ due to Bradač, and Malec and Tompkins, independently.

💡 Research Summary

This paper makes a significant contribution to spectral graph theory by generalizing a fundamental spectral Turán-type inequality to the setting of weighted graphs and deriving a wide array of consequences that unify and extend numerous classical results.

The starting point is the well-known spectral Turán theorem: for a (K_{r+1})-free graph (G) with (m) edges and spectral radius (\lambda(G)), Nikiforov proved (\lambda(G)^2 \leq 2(1-1/r)m). This was later refined by Liu and Ning to a local version: (\lambda(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\mathrm{cl}(e)-1}{\mathrm{cl}(e)}), where (\mathrm{cl}(e)) is the order of the largest clique containing edge (e).

The authors’ central result, Theorem 1.1, extends this local inequality to weighted graphs. They prove that for a weighted graph (G) with edge weight function (w), the inequality becomes (\lambda(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\mathrm{cl}(e)-1}{\mathrm{cl}(e)} w(e)^2). Furthermore, they provide a complete characterization of the extremal graphs for which equality holds: up to isolated vertices, they are complete multipartite graphs, and there exists a specific weight vector satisfying precise norm conditions related to the graph’s partition.

The power of this main theorem lies in its unifying and generative capacity. Its implications span two main domains:

1. Extremal Graph Theory: It provides a concise new proof for a localized Turán theorem independently discovered by Bradač, and Malec and Tompkins (Corollary 2.1). It also leads to new complementary lower bounds (Theorem 1.2), such as (\sum_{e \in E(G)} \frac{2}{\mathrm{cl}(e)} \geq n-1), and a novel vertex-degree-based inequality (Theorem 1.3). A key consequence is a localized version of Wilf’s theorem (Theorem 1.4, Corollary 2.7): (\lambda(G) \leq \sum_{v \in V(G)} \frac{\mathrm{cl}(v)-1}{\mathrm{cl}(v)}). This result independently confirms a conjecture posed by Tripathi originating from problems in Probability and Operator Algebras concerning Khintchine-type inequalities for mixtures of free and classical random variables.
2. Spectral Graph Theory: The main theorem serves as a common generalization of several classical spectral upper bounds. Its corollaries include Stanley’s inequality ((\lambda(G) \leq \sqrt{2m + 1/4} - 1/2)), Hong’s inequality ((\lambda(G) \leq \sqrt{2m - n + 1}) for (\delta(G)\ge 1)), and Nikiforov’s spectral Turán theorem. Notably, while Nikiforov’s inequality implied Stanley’s, it did not imply Hong’s—a gap now bridged by this work. The theorem also yields weighted generalizations of the Edwards-Elphick theorem (Corollary 2.8) and the Cvetković theorem (Corollary 2.9).

The proof of Theorem 1.1 relies on a novel weighted extension of the Motzkin-Straus theorem, optimized using Lagrange multipliers over the simplex, with the local clique size (\mathrm{cl}(v)) playing a crucial role. The paper demonstrates how the interplay between local graph structure (clique sizes per edge/vertex) and spectral radius can be captured in a single, powerful inequality applicable to weighted graphs. By doing so, it creates a unified framework that recovers, explains, and extends diverse results from both spectral and extremal graph theory, highlighting deep connections between local combinatorial parameters and global spectral properties.