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# 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

#### 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.

Dedicated to Professor Fan Chung, with admiration and thanks.

This work is supported by the National Natural Science Foundation of China (Nos. 12101166, 12101165, 11971311, 12161141003), Hainan Provincial Natural Science Foundation of China (Nos. 120RC453, 120MS002), Montenegrin-Chinese Science and Technology Cooperation Project (No. 3-12), and the Science and Technology Commission of Shanghai Municipality (No. 22JC1403602).

Received 18 March 2021

Received revised 11 March 2022

Accepted 28 March 2022

Published 29 March 2023