An Aα-spectral radius for a spanning tree with constrained leaf distance in a graph

Let ? ? [0, 1) be a real number, and let G be a connected graph of order n with n ? ?(?), where ?(?) = 9 for 0 ? ? ? 2/3 and ?(?) = 4/1?? for 2/3 < ? < 1. A spanning tree T of G is a subgraph of G that is a tree covers all vertices of G. The leaf distance of a tree is the minimum of distances between any two leaves of a tree. Let A?(G) = ?D(G)+(1??)A(G), whereA(G) is the adjacency matrix ofGandD(G) is the diagonal matrix of vertex degrees of G. The largest eigenvalues of A?(G), denoted by ??(G), is called A?-spectral radius of G. In this paper, it is proved that G has a spanning tree with leaf distance at least 4 if ??(G) ? ?(n), where ?(n) is the largest root of x3 ?(?n+n+??3)x2 +(?n2 +?2n??n?n?2?+1)x??2n2 +3?2n??n+n?4?2 +5??3 = 0.