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October 22, 2025Open Access

A Spectral Version of the Theorem of Zykov and Erdős

Key Points

A spectral version of the theorem by zykov and erdős reveals new graph properties.
The result generalizes nikiforov's theorem, indicating a broader application in graph theory.
This work extends previous findings of liu and bu, showcasing advancements in spectral graph theory.
Maximal copies of k_s within k_r-free graphs were uniquely identified through a new spectral approach.

Abstract

Zykov and Erdős showed independently that for 2 s r, the maximum number of copies of Kₛ among all Kᵣ-free n-vertex graphs is achieved uniquely on the complete balanced r-partite n-vertex graph (Turán graph T₍, ₑ). When s=2, it is the classical theorem of Turán. Nikiforov proved a spectral version of Turán's Theorem. In this paper, we give a spectral version of the theorem by Zykov and Erdős. Our result is a generalization of Nikiforov's Theorem and a theorem of Liu and Bu.

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