Characterization of trees with second minimum eccentricity energy

The eccentricity matrix of a connected graph G, denoted by epsilon(G), is obtained from the distance matrix of G by keeping the largest entries in each row and each column, and putting the remaining entries as zero. The eigenvalues of epsilon(G) are the epsilon-eigenvalues of G. The eccentricity energy (or the epsilon-energy) of G is the sum of the absolute values of all epsilon-eigenvalues of G. In this article, we characterize the trees with second minimum E-energy among all trees on n >= 5 vertices. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.