On the minimal energy of trees with a given diameter

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. F. Zhang, H. Li [On acyclic conjugated molecules with minimal energies, Discrete Appl. Math. 92 (1999) 71-84] characterized the trees with a perfect matching having the minimal and the second minimal energies, which solved a conjecture proposed by I. Gutman [Acyclic conjugated molecules, trees and their energies, J. Math. Chem. 1 (1987) 123-143]. In this letter, for a given positive integer d we characterize the tree with the minimal energy having diameter at least d. As a corollary, we also characterize the tree with the minimal Hosoya index having diameter at least d.