Characterization of self-injective Leavitt path algebras
A Leavitt path algebra $L_K(E)$ is left (right) self-injective if and only if $E$ is a row-finite acyclic graph in which every infinite path contains a line point and so, if and only if $L_K(E)$ is a semisimple ring.
Reference(s)
G. A. Pino, K. M. Rangaswamy, and M. S. Molina. Weakly regular and self-injective Leavitt path algebras over arbitrary graphs. (2011) @ Section 4