  The eccentricity (anti-adjacency) matrix $\varepsilon(G)$ of a graph $G$ is
obtained from the distance matrix by retaining the eccentricities in each row
and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its
eccentricity matrix $\varepsilon(G),$ and the eccentricity energy (or the
$\varepsilon$-energy) of $G$ is the sum of the absolute values of
$\varepsilon$-eigenvalues. In this paper, we establish some bounds for the
$\varepsilon$-energy of the complete multipartite graph $K_{n_1, n_2,
\ldots,n_p}$ of order $n= \sum_{i=1}^p n_i $ and characterize the extreme
graphs.
  This partially answers the problem given in Wang {\em et al.} (2019). We
finish the paper showing graphs that are not $\varepsilon$-cospectral with the
same $\varepsilon$-energy.
