D-index and Q-index for spanning trees with leaf degree at most k 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 is an element of 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 D(G) and Q(G) denote the distance matrix and the distance signless Laplacian matrix of G, respectively. In this work, we provide the upper bounds for the spectral radius of D(G) (resp. Q(G)) in a connected graph G of order n to guarantee that G contains a spanning tree with leaf degree at most k. Furthermore, we establish some extremal graphs to show all the upper bounds obtained in this work are sharp. (c) 2024 Elsevier B.V. All rights reserved.