NEW EXTREMAL RESULTS FOR THE DISTANCE LAPLACIAN SPECTRAL RADIUS OF TREES

The distance Laplacian matrix of a graph G is defined as L(G) = Tr(G)− D(G), where Tr(G) and D(G) are, respectively, the diagonal matrix of vertex transmissions and the distance matrix of G. Inside the set Tn of trees with n vertices, we consider the subsets NCn and NSn containing non-caterpillar trees and non-starlike trees respectively, and study the graphs with maximum distance Laplacian spectral radii in NCn, in NSn, and in NCn ∩ NSn. As a by-product, we pick out the three candidates to attain the fourth biggest maximum Laplacian spectral radius in Tn