A survey of edge-spectral-Tur\'an type problems in spectral graph theory: Results, conjectures and open problems

TL;DR

This survey reviews recent advances, conjectures, and open problems in edge-spectral-Turán problems within spectral graph theory, focusing on maximizing spectral radius in F-free graphs with a given number of edges.

Contribution

It compiles and discusses key results, conjectures, and open questions related to edge-spectral-Turán problems, providing a comprehensive overview for future research.

Findings

Summarizes recent results in spectral Turán problems.

Proposes new conjectures for spectral radius bounds.

Identifies open problems and directions for further study.

Abstract

The edge-spectral-Tur\'an type problem is also called the Brualdi-Hoffman-Tur\'an type problem, which is a central topic in spectral graph theory, seeking to determine the maximum spectral radius $\lambda(G)$ 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.