Spectral extremal results on the $A_\alpha$-spectral radius of graphs without $K_{a,b}$-minor

TL;DR

This paper reviews and extends recent spectral Turán results related to $K_{a,b}$-minor free graphs, providing a comprehensive understanding of the $A_{\alpha}$-spectral radius in extremal graph theory.

Contribution

It fully states, generalizes, and provides a self-contained proof of the spectral Turán theorem for $K_{a,b}$-minor free graphs, enhancing understanding of their spectral properties.

Findings

Complete statement and proof of the spectral Turán theorem for $K_{a,b}$-minor free graphs.

Generalization of previous results with a self-contained proof.

Deeper insight into the relationship between $A_{\alpha}$-spectral radius and extremal graph structure.

Abstract

An important theorem about the spectral Tur\'an problem of $K_{a,b}$ was largely developed in separate papers. Recently it was completely resolved by Zhai and Lin [J. Comb. Theory, Ser. B 157 (2022) 184-215], which also confirms a conjecture proposed by Tait [J. Comb. Theory, Ser. A 166 (2019) 42-58]. Here, the prior work is fully stated, and then generalized with a self-contained proof. The more complete result is then used to better understand the relationship between the $A_{\alpha}$-spectral radius and the structure of the corresponding extremal $K_{a,b}$-minor free graph.