A generalization on spectral extrema of $K_{s,t}$-minor free graphs

TL;DR

This paper extends spectral extremal graph theory by determining the maximum $A_eta$-spectral radius for $K_{s,t}$-minor free graphs, generalizing previous adjacency spectral results and solving a related conjecture.

Contribution

It generalizes the spectral extremal problem to the matrix $A_eta(G)$ for $0<eta<1$, and characterizes extremal graphs, also solving a conjecture in the field.

Findings

Determined the extremal graph for maximum $A_eta$-spectral radius among $K_{s,t}$-minor free graphs.

Extended spectral extremal results from adjacency matrices to the $A_eta$ matrix.

Solved a conjecture related to spectral extremal problems.

Abstract

The spectral extrema problems on forbidding minors have aroused wide attention. Very recently, Zhai and Lin [J. Combin. Theory Ser. B 157 (2022) 184--215] determined the extremal graph with maximum adjacency spectral radius among all $K_{s,t}$-minor free graphs of sufficiently large order. The matrix $A_{\alpha}(G)$ is a generalization of the adjacency matrix $A(G)$, which is defined by Nikiforov \cite{Nikiforov2} as $$A_{\alpha}(G) = \alpha D(G) + (1 - \alpha)A(G),$$ where $0\leq\alpha \leq1$. Given a graph $F$, the $A_\alpha$-spectral extrema problem is to determine the maximum spectral radius of $A_{\alpha}(G)$ or characterize the extremal graph among all graphs with no subgraph isomorphic to $F$. For $\alpha=0$, the matrix $A_{\alpha}(G)$ is exactly the adjacency matrix $A(G)$. Motivated by the nice work of Zhai and Lin, in this paper we determine the extremal graph with maximum $A_\alpha$-spectral radius among all $K_{s,t}$-minor free graphs of sufficiently large order, where $0<\alpha<1$ and $2\leq s\leq t$. As by-products, we completely solve the Conjecture posed by Chen and Zhang in [Linear Multilinear Algebra 69 (10) (2021) 1922--1934].