Determining the conjugated trees with the third-through the sixth-minimal energies

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. A conjugated tree is a tree that has a perfect matching. The conjugated trees F_n and B _n with the minimal and secondminimal energies were determined by Zhang and Li. They also figured out that the conjugated trees with the third- and fourth-minimal energies are L_n or M_n. However, they could not determine which is the third, and the other is the fourth. Recently, S. Li and N. Li further investigated the conjugated trees with the third-, through the sixth-minimal energies. As a result, they figured out that these trees must be among the trees L_n, M_n, I_n and W_n/2 ^*. They then showed that the energy of M _n is smaller than that of I_n, and the energy of L _n is smaller than that of W_n/2 ^*, but they could not give a total ordering of the 4 trees. For comparing of the energies, a common used method is to compare the number of fc-matchings in each concerned tree, but it is often invalid for further comparing. This paper is aimed at solving the above unsolved problems by giving the energies of the 4 trees a total ordering, that is, completely determining the conjugated trees with the third-, forth-, fifth- and sixth-minimal energies. Our method uses the well-known Coulson integral formula.