The effect of graft transformations on distance signless Laplacian spectral radius

For a connected graph G, the distance signless Laplacian spectral radius of G is the spectral radius of its distance signless Laplacian matrix Q(G) defined as Q(G)=Tr(G)+D(G), where Tr(G) is the diagonal matrix of vertex transmissions of G and D(G) is the distance matrix of G. In this paper, we study the effect of three types of graft transformations to decrease and/or increase the distance signless Laplacian spectral radius. As applications, on one hand, we determine the unique graphs with minimum distance signless Laplacian spectral radius among non-starlike trees, among non-caterpillar trees, and among non-starlike non-caterpillar trees, respectively, and determine the unique trees with third and forth minimum distance signless Laplacian spectral radius, respectively, and on the other hand, we determine the unique graphs with maximum distance signless Laplacian spectral radius among trees with given maximum degree, among trees of a perfect matching with maximum degree, among non-starlike trees, among non-caterpillar trees, among non-starlike non-caterpillar trees, and among unicyclic odd-cycle graphs, respectively, and determine the unique trees with second and third maximum distance signless Laplacian spectral radius, respectively, and the unique connected graph with second maximum distance signless Laplacian spectral radius.