Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 6

Special issue in honor of Fan Chung

Guest editors: Paul Horn, Yong Lin, and Linyuan Lu

Spectral extremal results on the $\alpha$-index of graphs without minors and star forests

Pages: 2355 – 2378

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n6.a2

Authors

Ming-Zhu Chen (School of Science, Hainan University, Haikou, China)

A-Ming Liu (School of Science, Hainan University, Haikou, China)

Xiao-Dong Zhang (Shanghai Jiao Tong University, Shanghai, China)

Abstract

Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_\alpha (G)$ of $A(G)$ and $D(G)$ by $A_\alpha (G) = \alpha D(G)+(1-\alpha )A(G)$ for any real number $0 \leq \alpha \leq 1$. The $\alpha $-index of $G$ is the largest eigenvalue of $A_\alpha (G)$. In this paper, using some new eigenvector techniques introduced by Tait and coworkers, we determine the maximum $\alpha$-index and characterize all extremal graphs for $K_r$ minor-free graphs, $K_{s,t}$ minor-free graphs, and starforest-free graphs for any $0 \lt \alpha \lt 1$ respectively.

Keywords

spectral radius, $\alpha$-index, extremal graphs, star forests, minors

2010 Mathematics Subject Classification

Primary 05C50, 05C83. Secondary 05C35.

Received 18 March 2021

Received revised 11 March 2022

Accepted 28 March 2022

Published 29 March 2023