On the two largest distance eigenvalues of graph powers

We give sharp lower bounds for the largest and the second largest distance eigenvalues of the k-th power of a connected graph, determine all trees and unicyclic graphs for which the second largest distance eigenvalues of the squares are less than 5−32, and determine the unique n-vertex trees of which the squares achieve minimum and second-minimum largest distance eigenvalues, as well as the unique n-vertex trees of which the squares achieve minimum, second-minimum and third-minimum second largest distance eigenvalues.