论文标题
$ k_ {s，t} $的光谱极端概括 - 次要免费图形
A generalization on spectral extrema of $K_{s,t}$-minor free graphs
论文作者
论文摘要
禁止未成年人的极端问题引起了广泛关注。最近，Zhai和Lin [J.组合。理论ser。 B 157（2022）184--215]确定了所有$ k_ {s，t} $之间具有最大邻接光谱半径的极端图 - 足够大订单的次要免费图形。矩阵$a_α（g）$是邻接矩阵$ a（g）$的概括，由nikiforov \ cite {nikiforov2}定义为$$a_α（g）=αd（g）=αd（g） +（g） +（1-α）a（g），$ 0 \ $ 0 \ 0 \ 0 \ leq leqleqα\ leq leq leq leq leq leq leq1 $。给定图形$ f $，$a_α$ spectral极点问题是确定$a_α（g）$的最大光谱半径，或在所有图中表征极端图中的极端图，而没有子图等构象为$ f $。对于$α= 0 $，矩阵$a_α（g）$正是邻接矩阵$ a（g）$。由Zhai和Lin的出色工作激励，在本文中，我们确定了所有$ k_ {s，t} $中最大$a_α$ spectral半径的极端图，其中$ 0 <α<1 $ and $ 2 \ leq s \ leq s \ leq t $ $ 0 <α<1 $ and $ 0 <α<1 $。作为副产品，我们在[线性多线性代数69（10）（2021）1922--1934]中完全解决了Chen和Zhang提出的猜想。
The spectral extrema problems on forbidding minors have aroused wide attention. Very recently, Zhai and Lin [J. Combin. Theory Ser. B 157 (2022) 184--215] determined the extremal graph with maximum adjacency spectral radius among all $K_{s,t}$-minor free graphs of sufficiently large order. The matrix $A_α(G)$ is a generalization of the adjacency matrix $A(G)$, which is defined by Nikiforov \cite{Nikiforov2} as $$A_α(G) = αD(G) + (1 - α)A(G),$$ where $0\leqα\leq1$. Given a graph $F$, the $A_α$-spectral extrema problem is to determine the maximum spectral radius of $A_α(G)$ or characterize the extremal graph among all graphs with no subgraph isomorphic to $F$. For $α=0$, the matrix $A_α(G)$ 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 $A_α$-spectral radius among all $K_{s,t}$-minor free graphs of sufficiently large order, where $0<α<1$ 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].
