Minimizers for the energy of eccentricity matrices of trees

The eccentricity matrix of a connected graph G, denoted by E(G), is constructed from the distance matrix of G by keeping only the largest nonzero elements in each row and each column and setting the remaining entries as zero. The E-eigenvalues of G are the eigenvalues of E(G). The eccentricity energy (or the E-energy) of G is the sum of the absolute values of all E-eigenvalues of G. In this article, we determine the unique tree whose second largest E-eigenvalue is minimum among all trees on n vertices other than the star. Then, we characterize the trees with minimum E-energy among all trees on n vertices.