Minimal energies of trees with given parameters

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Let T(k) be the set of trees with given order k. Suppose that T∈T(k) and {v1,v2,…,vk} be the ordering vertex set of T. We denote by T(n1,n2,…,nk) the graph obtained by attaching ni pendent vertices to vertex of T respectively. Let T(n,k)={T(n1,n2,…,nk)|T∈T(k),n1+n2+⋯+nk=n-k,ni⩾1,i=1,2,…,k}. In this paper, we determine the trees in T(n,k) with the first and the second minimal energies. As applications, we can characterize the trees with the first and the second minimal energies among the set of trees with given domination number, matching number, independence number respectively.