Convex and quasiconvex functions on trees and their applications

We introduce convex and quasiconvex functions on trees and prove that for a tree the eccentricity, transmission and weight functions are strictly quasiconvex. It is shown that the Perron vector of the distance matrix is strictly convex whereas the Perron vector of the distance signless Laplacian is quasiconvex for a tree. In the class of all trees with a given number of pendent vertices, we prove that the distance Laplacian and distance signless Laplacian spectral radius are both maximized at a dumbbell. Among all trees with fixed maximum degree, we prove that the broom is the unique tree that maximizes the distance Laplacian and distance signless Laplacian spectral radius. We find the unique graph that maximizes the distance spectral radius in the class of all unicyclic graphs of girth g on n vertices. Also we find the unique graph that maximizes the distance signless Laplacian and the distance Laplacian spectral radius in the class of all unicyclic graphs on n vertices.