Eccentric graph of trees and their Cartesian products

Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph, $E(G)$ of $G$ is a graph defined on the same vertex set as of $G$ and two vertices are adjacent if one is eccentric to the other. We find the structure and the girth of the eccentric graph of trees and see that the girth of the eccentric graph of a tree can either be zero, three, or four. Further, we study the structure of the eccentric graph of the Cartesian product of graphs and prove that the girth of the eccentric graph of the Cartesian product of trees can only be zero, three, four or six. Furthermore, we provide a comprehensive classification when the eccentric girth assumes these values. We also give the structure of the eccentric graph of the grid graphs and the Cartesian product of cycles. Finally, we determine the conditions under which the eccentricity matrix of the Cartesian product of trees becomes invertible.