The minimum ABC energy of trees

Let G be a graph of order n, and di be the degree of its i-th vertex. The ABC matrix of G is the matrix of order n whose (i,j)-entry is equal to (di+dj−2)/(didj) if the i-th vertex and the j-th vertex of G are adjacent, and 0 otherwise. The ABC eigenvalues of G are the eigenvalues of its ABC matrix, and the ABC energy of G is the sum of the absolute values of its ABC eigenvalues. In Chen (2018) [9], the author conjectured that the star has the minimum ABC energy among all trees. In this paper, we prove the conjecture is true.