For real $\alpha\in [0,1)$ and a hypergraph $G$, the $\alpha$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ is the adjacency matrix of $G$, which is a symmetric matrix with zero diagonal such that for distinct vertices $u,v$ of $G$, the $(u,v)$-entry of $A(G)$ is exactly the number of edges containing both $u$ and $v$, and $D(G)$ is the diagonal matrix of row sums of $A(G)$. We study the $\alpha$-spectral radius of a hypergraph that is uniform or not necessarily uniform. We propose some local grafting operations that increase or decrease the $\alpha$-spectral radius of a hypergraph. We determine the unique hypergraphs with maximum $\alpha$-spectral radius among $k$-uniform hypertrees, among $k$-uniform unicyclic hypergraphs, and among $k$-uniform hypergraphs with fixed number of pendant edges. We also determine the unique hypertrees with maximum $\alpha$-spectral radius among hypertrees with given number of vertices and edges, the unique hypertrees with the first three largest (two smallest, respectively) $\alpha$-spectral radii among hypertrees with given number of vertices, the unique hypertrees with minimum $\alpha$-spectral radius among the hypertrees that are not $2$-uniform, the unique hypergraphs with the first two largest (smallest, respectively) $\alpha$-spectral radii among unicyclic hypergraphs with given number of vertices, and the unique hypergraphs with maximum $\alpha$-spectral radius among hypergraphs with fixed number of pendant edges.

翻译：对于实数 $\alpha\in [0,1)$ 和超图 $G$，$G$ 的 $\alpha$-谱半径定义为矩阵 $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ 的最大特征值，其中 $A(G)$ 是 $G$ 的邻接矩阵（对称矩阵，对角元为零，且对于 $G$ 中不同顶点 $u,v$，$A(G)$ 的 $(u,v)$ 元恰好等于同时包含 $u$ 和 $v$ 的边的数目），$D(G)$ 是以 $A(G)$ 的行和为对角元的对角矩阵。我们研究了均匀或非均匀超图的 $\alpha$-谱半径。提出了一些能增加或减少超图 $\alpha$-谱半径的局部嫁接操作。确定了在 $k$-均匀超树、$k$-均匀单圈超图以及具有固定悬挂边数的 $k$-均匀超图中，具有最大 $\alpha$-谱半径的唯一超图。我们还确定了在给定顶点数和边数的超树中具有最大 $\alpha$-谱半径的唯一超树，在给定顶点数的超树中具有前三最大（分别地，前两最小）$\alpha$-谱半径的唯一超树，在非 $2$-均匀超树中具有最小 $\alpha$-谱半径的唯一超树，在给定顶点数的单圈超图中具有前两最大（分别地，最小）$\alpha$-谱半径的唯一超图，以及在具有固定悬挂边数的超图中具有最大 $\alpha$-谱半径的唯一超图。