On the $\alpha$-spectral radius of the $k$-uniform supertrees

Let $G$ be a $k$-uniform hypergraph with vertex set $V(G)$ and edge set $E(G)$. A connected and acyclic hypergraph is called a supertree. For $0\leq\alpha<1$, the $\alpha$-spectral radius of $G$ is the largest $H$-eigenvalue of $\alpha D(G)+(1-\alpha)A(G)$, where $D(G)$ and $A(G)$ are the diagonal tensor of the degrees and the adjacency tensor of $G$, respectively. In this paper, we determine the unique supertrees with the maximum $\alpha$-spectral radius among all $k$-uniform supertrees with $m$ edges and independence number $\beta$ for $\lceil\frac{m(k-1)+1}{k}\rceil\leq\beta\leq m$, among all $k$-uniform supertrees with given degree sequences, and among all $k$-uniform supertrees with $m$ edges and matching number $\mu$ for $1\leq\mu\leq\lfloor\frac{m(k-1)+1}{k}\rfloor$, respectively.