Abstract:
Let $G$ be a connected graph of order $n$. For any integer $k\geq2$, a spanning $k$-tree of $G$ is a spanning tree in which every vertex has degree at most $k$. In this paper, we provide a tight $A_{\alpha}$-spectral condition to guarantee the existence of a spanning $k$-tree in $G$ with extremal graphs being characterized. Moreover, we also present tight $A_{\alpha}$-spectral conditions for $G$ admitting a spanning $k$-ended-tree (i.e., a spanning tree with at most $k$ leaves) and determine the extremal graphs.
