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Localization of the clique spectral version of Zykov's theorem

Changjiang Bu, Jueru Liu, Haotian Zeng

Abstract

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.

Localization of the clique spectral version of Zykov's theorem

Abstract

Zykov's theorem shows that -partite Turán graph uniquely has the maximum number of among all -vertex -free graphs for . 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.

Paper Structure

This paper contains 3 sections, 17 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Theorem 1.1

nikiforov2002 Let $G$ be an $n$-vertex graph with clique number $\omega$. Then Equality holds if and only if $G$ is a complete bipartite graph for $\omega=2$, or a complete regular $\omega$-partite graph for $\omega\ge3$ and $\omega$ divides $n$ (possibly with some isolated vertices).

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • ...and 14 more