Localization of the clique spectral version of Zykov's theorem

Zykov’s theorem shows that $r$-partite Turán graph uniquely has the maximum number of $K_t$ among all $n$-vertex $K_{r+1}$-free graphs for $2\le t\le r$. The clique tensor is a high-order extension of the adjacency matrix of a graph. Yu and Peng \cite{peng1} gave a spectral version of the Zykov’s theorem via clique tensor. In this paper, we give some upper bounds on the spectral radius of the clique tensor of a graph, which can be viewed as the localizations of the spectral version of Zykov’s theorem.