Localization of the clique spectral version of Zykov's theorem

TL;DR

This paper extends Zykov's theorem using clique tensors, providing upper bounds on their spectral radius, thus localizing the spectral version of the theorem.

Contribution

It introduces bounds on the spectral radius of the clique tensor, offering a localized spectral perspective of Zykov's theorem.

Findings

Provided upper bounds on the spectral radius of the clique tensor.

Localized the spectral version of Zykov's theorem using clique tensors.

Extended the spectral analysis of Turán graphs to high-order tensors.

Abstract

Zykov's theorem shows that $r$-partite Tur\'{a}n 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.