The changes in distance Laplacian spectral radius of graphs resulting from graft transformations

For a connected graph G, the distance Laplacian spectral radius of G is the spectral radius of its distance Laplacian matrix L(G) defined as L(G)=Tr(G)−D(G), where Tr(G) is a diagonal matrix of vertex transmissions of G and D(G) is the distance matrix of G. In this paper, we study the change in the distance Laplacian spectral radius of graphs by some graft transformations, and as applications, we determine the unique graphs with minimum distance Laplacian spectral radius among non-caterpillar trees, and among non-starlike non-caterpillar trees, respectively, we prove that the path is the unique graph with maximum distance Laplacian spectral radius among connected graphs, and determine the unique graph with maximum distance Laplacian spectral radius among connected graphs with given clique number.