Spanning trees and signless Laplacian spectral radius in graphs

Let G be a connected graph and let k be a positive integer. Let T be a spanning tree of G. The leaf degree of a vertex v V (T) is defined as the number of leaves adjacent to v in T. The leaf degree of T is the maximum leaf degree among all the vertices of T. Let A (G) be the adjacency matrix of G and D (G) be the diagonal degree matrix of G. Let Q (G) =D (G) +A (G) be the signless Laplacian matrix of G. The largest eigenvalue of Q (G), denoted by q (G), is called the signless Laplacian spectral radius of G. In this paper, we investigate the connection between the spanning tree and the signless Laplacian spectral radius of G, and put forward a sufficient condition based upon the signless Laplacian spectral radius to guarantee that a graph G contains a spanning tree with leaf degree at most k. Finally, we construct some extremal graphs to claim all the bounds obtained in this paper are sharp.