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 Tn,p,Tn,p,Tn,dTn,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 Tn,pTn,p (Tn,d,Tn,d, respectively) for 4≤p≤n−94≤p≤n−9 (10≤d≤n−3,10≤d≤n−3, respectively), which solves the problems left in Ma (2014).