The proof of a conjecture on the comparison of the energies of trees

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the graph. In this paper, we first present a new method to directly compare the energies of two bipartite graphs, then also present some new techniques to compare the quasi-orders of some bipartite graphs. As the applications of these methods, we prove that a conjecture proposed by Wang and Kang (J Math Chem 47(3):937-958, 2010) is true. At the same time, our results also provide the simplified proofs of the main results of Wang and Kang (J Math Chem 47(3):937-958, 2010) and Li and Li (Electron J Linear Algebra 17:414-425, 2008).