Spectral radius and spanning trees of graphs

For integer $k\ge 2,$ a spanning $k$-ended-tree is a spanning tree with at most $k$ leaves. Motivated by the closure theorem of Broersma and Tuinstra [Independence trees and Hamilton cycles, J. Graph Theory 29 (1998) 227–237], we provide tight spectral conditions to guarantee the existence of a spanning $k$-ended-tree in a connected graph of order $n$ with extremal graphs being characterized. Moreover, by adopting Kaneko’s theorem [Spanning trees with constraints on the leaf degree, Discrete Appl. Math. 115 (2001) 73–76], we also present tight spectral conditions for the existence of a spanning tree with leaf degree at most $k$ in a connected graph of order $n$ with extremal graphs being determined, where $k\ge 1$ is an integer.