Characterization of trees with second minimum eccentricity energy

Abstract

The eccentricity matrix of a connected graph $G$, denoted by $E\left(G\right)$, 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 $E\left(G\right)$ are the $E$-eigenvalues of $G$. 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 characterize the trees with second minimum $E$-energy among all trees on $n\ge 5$ vertices.