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A survey of edge-spectral-Turán type problems in spectral graph theory: Results, conjectures and open problems

Yuantian Yu, Huihui Zhang, Minjie Zhang

TL;DR

This survey consolidates the edge-spectral-Turán problem in spectral graph theory, focusing on maximizing the spectral radius among F-free graphs with m edges. It organizes results into three forbidding regimes—consecutive odd cycles, subgraphs with χ ≤ 3, and color-critical subgraphs with χ ≥ 4—highlighting key extremal graph structures such as blow-ups, joins, and complete multipartite graphs. The paper synthesizes exact bounds, asymptotic results, and numerous conjectures, providing a comprehensive resource for understanding how forbidden subgraph structures shape spectral extremality. Overall, it frames a cohesive picture of current knowledge and open questions, guiding future research in spectral extremal graph theory.

Abstract

The edge-spectral-Turán type problem is also called the Brualdi-Hoffman-Turán type problem, which is a central topic in spectral graph theory, seeking to determine the maximum spectral radius $λ(G)$ of an $F$-free graph $G$ with $m$ edges. This problem has attracted significant attention in recent years. In this paper, we will sort out several closely related results in this type of problem and then propose some conjectures for further research.

A survey of edge-spectral-Turán type problems in spectral graph theory: Results, conjectures and open problems

TL;DR

This survey consolidates the edge-spectral-Turán problem in spectral graph theory, focusing on maximizing the spectral radius among F-free graphs with m edges. It organizes results into three forbidding regimes—consecutive odd cycles, subgraphs with χ ≤ 3, and color-critical subgraphs with χ ≥ 4—highlighting key extremal graph structures such as blow-ups, joins, and complete multipartite graphs. The paper synthesizes exact bounds, asymptotic results, and numerous conjectures, providing a comprehensive resource for understanding how forbidden subgraph structures shape spectral extremality. Overall, it frames a cohesive picture of current knowledge and open questions, guiding future research in spectral extremal graph theory.

Abstract

The edge-spectral-Turán type problem is also called the Brualdi-Hoffman-Turán type problem, which is a central topic in spectral graph theory, seeking to determine the maximum spectral radius of an -free graph with edges. This problem has attracted significant attention in recent years. In this paper, we will sort out several closely related results in this type of problem and then propose some conjectures for further research.
Paper Structure (4 sections, 52 theorems, 10 equations)

This paper contains 4 sections, 52 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Forbidding subgraphs involving consecutive odd cycles
  3. Forbidding a subgraph whose chromatic number is at most 3
  4. Forbidding a color-critical subgraph whose chromatic number is at least 4

Key Result

Theorem 2.1

For all graphs $G \in \mathcal{G}(m, C_3),$ one has $\lambda(G) \leqslant \sqrt{m}$.

Theorems & Definitions (68)

  • Theorem 2.1: Nosal Nosal, 1970
  • Theorem 2.2: Nikiforov VNK2009, 2009
  • Theorem 2.3: Lin, Ning and Wu 9, 2021
  • Theorem 2.4: Lin, Ning and Wu 9, 2021
  • Theorem 2.5: Zhai and Shu 26, 2022
  • Theorem 2.6: Li, Zhao and Shen LZS, 2025+
  • Theorem 2.7: Sun and Li sl2023, 2023
  • Theorem 2.8: Li, Peng Li-Peng, Sun, Li sl2023
  • Theorem 2.9: Li and Yu LY2023, 2023
  • Theorem 2.10: Lou, Lu and Huang Lou-Lu-Huang, 2024
  • ...and 58 more