Minimizers for the energy of eccentricity matrices of trees

Abstract

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.