  The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the
distance matrix by retaining the eccentricities (the largest distance) in each
row and each column. In this paper, we give a characterization of the star
graph, among the trees, in terms of invertibility of the associated
eccentricity matrix. The largest eigenvalue of $\varepsilon(G)$ is called the
$\varepsilon$-spectral radius, and the eccentricity energy (or the
$\varepsilon$-energy) of $G$ is the sum of the absolute values of the
eigenvalues of $\varepsilon(G)$. We establish some bounds for the
$\varepsilon$-spectral radius and characterize the extreme graphs. Two graphs
are said to be $\varepsilon$-equienergetic if they have the same
$\varepsilon$-energy. For any $n \geq 5$, we construct a pair of
$\varepsilon$-equienergetic graphs on $n$ vertices, which are not
$\varepsilon$-cospectral.
