A Spectral Version of the Theorem of Zykov and Erdős

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 Tn,r). 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.