Solutions to unsolved problems on the minimal energies of two classes of trees

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Let T n , p , T n , d be the set of all trees of order n with p pendent vertices, diameter d, respectively. In this paper, we completely characterize the trees with second-minimal and third-minimal energy in T n , p ( T n , d , respectively) for 4 ? p ? n - 9 ( 10 ? d ? n - 3 , respectively), which solves the problems left in Ma (2014).