A Spectral Version of the Theorem of Zykov and Erdős

Abstract

Zykov and Erdős showed independently that for $2\le s\le r$, the maximum number of copies of $K_s$ among all $K_r$-free $n$-vertex graphs is achieved uniquely on the complete balanced $r$-partite $n$-vertex graph (Turán graph $T_{n,r}$). 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.