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Turán extremal graphs vs. Signless Laplacian spectral Turán extremal graphs

Ming-Zhu Chen, Ya-Lei Jin, Peng-Li Zhang, Xiao-Dong Zhang

TL;DR

The paper addresses when the signless Laplacian spectral Turán extremal graphs must coincide with classical Turán-extremal graphs for $F$-free graphs, under the regime $ ext{ex}(n,F)=t_r(n)+O(1)$ with $r\ge 3$. It combines Szemerédi's regularity lemma with Füredi's stability to derive a precise near-Turán structural description for large $n$, and then uses Perron-vector analysis of the signless Laplacian to show that any $F$-free graph maximizing $q(G)$ must be a Turán graph (up to $O(1)$ edits). Consequently, $Ex_{ssp}(n,F)\subseteq Ex(n,F)$ holds for sufficiently large $n$, meaning spectral extremality aligns with Turán extremality in this setting; the work also provides a negative answer to a related conjecture on the adjacency-spectral analogue. These results advance understanding in spectral Turán theory by clarifying when signless Laplacian optimization enforces classic Turán structure, and they delineate limitations via counterexamples in related regimes.

Abstract

Let $F$ be a graph with chromatic number $χ(F) = r+1$. Denote by $ex(n, F)$ and $Ex(n, F)$ the Turán number and the set of all extremal graphs for $F$, respectively. In addition, $ex_{ssp}(n, F)$ and $Ex_{ssp}(n, F)$ are the maximum signless Laplacian spectral radius of all $n$-vertex $F$-free graphs and the set of all $n$-vertex $F$-free graphs with signless Laplacian spectral radius $ex_{ssp}(n, F)$, respectively. It is known that $Ex_{ssp}(n, F)\supset Ex(n, F)$ if $F$ is a triangle. In this paper, employing the regularity method and Füredi's stability theorem, we prove that for a given graph $F$ and $r\geqslant 3$, if $ex(n, F) = t_r(n)+O(1)$, then $ Ex_{ssp}(n, F) \subseteq Ex(n, F)$ for sufficiently large $n$, where $t_r(n)$ is the number of edges in the Turán graph $T_r(n)$.

Turán extremal graphs vs. Signless Laplacian spectral Turán extremal graphs

TL;DR

The paper addresses when the signless Laplacian spectral Turán extremal graphs must coincide with classical Turán-extremal graphs for -free graphs, under the regime with . It combines Szemerédi's regularity lemma with Füredi's stability to derive a precise near-Turán structural description for large , and then uses Perron-vector analysis of the signless Laplacian to show that any -free graph maximizing must be a Turán graph (up to edits). Consequently, holds for sufficiently large , meaning spectral extremality aligns with Turán extremality in this setting; the work also provides a negative answer to a related conjecture on the adjacency-spectral analogue. These results advance understanding in spectral Turán theory by clarifying when signless Laplacian optimization enforces classic Turán structure, and they delineate limitations via counterexamples in related regimes.

Abstract

Let be a graph with chromatic number . Denote by and the Turán number and the set of all extremal graphs for , respectively. In addition, and are the maximum signless Laplacian spectral radius of all -vertex -free graphs and the set of all -vertex -free graphs with signless Laplacian spectral radius , respectively. It is known that if is a triangle. In this paper, employing the regularity method and Füredi's stability theorem, we prove that for a given graph and , if , then for sufficiently large , where is the number of edges in the Turán graph .
Paper Structure (4 sections, 17 theorems, 56 equations)

This paper contains 4 sections, 17 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The structure of graphs in $\mathcal{G}_{n}$
  4. Proof of Theorem \ref{['main result']}

Key Result

Theorem 1.1

Let $F$ be a color-critical graph with $\chi(F)=r+1$. Then there exists a number $n_0(F)$ such that $T_r(n)$ is the only extremal graph with respect to $ex(n, F)$ provided $n \geqslant n_0(F)$.

Theorems & Definitions (24)

  • Theorem 1.1: Simonovits, Simonovits1968
  • Theorem 1.2: Wang Kang Xue 2023
  • Theorem 1.4
  • Theorem 2.1: Szemerédi's graph regularity lemma, Zhao 2023
  • Theorem 2.2: Graph counting lemma, Zhao 2023
  • Lemma 2.3: Furedi2015
  • Theorem 2.4: Zheng Li Li 2025
  • Lemma 2.5: Cioaba Feng Tait Zhang 2020
  • Theorem 3.1
  • Claim 1
  • ...and 14 more