Extremal problems for the eccentricity matrices of complements of trees

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance $G$ keeping largest nonzero entries in each row and column leaving zeros remaining ones. $\mathcal{E}$-eigenvalues are eigenvalues $\mathcal{E}(G)$. modulus an eigenvalue $\mathcal{E}$-spectral radius $G$. $\mathcal{E}$-energy sum absolute values all In this article, we study some extremal problems for matrices complements trees characterize graphs. First, determine unique tree whose complement has minimum (respectively, maximum) among trees. Then, prove that symmetric about origin. As consequence these results, least Finally, discuss second $\mathcal{E}$-eigenvalue application, obtain Nordhaus-Gaddum-type lower bounds its complement.