Sufficient conditions for spanning trees with constrained leaf distance in a graph

The leaf distance of a tree is the minimum of distances between any two leaves of a tree. It is well known that seeking sufficient conditions for a graph to have some special kinds of spanning trees is an interesting and popular problem. In this paper, we first provide a lower bound on the size of a graph G to guarantee that G has a spanning tree with leaf distance at least 4. Moreover, for any graph G with minimum degree , we also deduce a lower bound on the spectral radius (or the signless Laplacian spectral radius) of G to ensure the existence of a spanning tree with leaf distance of at least 4 in G.