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. Zhang et al (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 Gutman (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.