Paper
A new spectral Turán theorem for weighted graphs and consequences
Authors
Lele Liu, Bo Ning
Abstract
Confirming a conjecture of Elphick and Edwards and strengthening a spectral theorem of Wilf, Nikiforov proved that for any -free graph , , where is the spectral radius of , and is the number of edges of . This result was later improved in \cite{LiuN26}, where it was shown that for any graph , , where denotes the order of the largest clique containing the edge . In this paper, we further extend this inequality to weighted graphs, proving that 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 , which complements an upper bound due to Bradač, and Malec and Tompkins, independently.