  We investigate conditions under which the endomorphism ring of the Leavitt
path algebra $L_{K}(E)$ possesses various ring and module-theoretical
properties such as being von Neumann regular, $\pi$-regular, strongly
$\pi$-regular or self-injective. We also describe conditions under which
$L_{K}(E)$ is continuous as well as automorphism invariant as a right
$L_{K}(E)$-module.
