Leavitt path algebras with bounded index of nilpotence and simple modules over them

In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence and show that each graded simple module $S$ over a Leavitt path algebra with bounded index of nilpotence is graded $\Sigma$-injective, that is, $S^{(\alpha)}$ is graded injective for any cardinal $\alpha$. Furthermore, we characterize Leavitt path algebras over which each simple module is $\Sigma$-injective. We have shown that each simple module over a Leavitt path algebra $L_K(E)$ is $\Sigma$-injective if and only if the graph $E$ contains no cycles, and there is a positive integer $d$ such that the length of any path in $E$ is less than or equal to $d$ and the number of distinct paths ending at any vertex $v$ (including $v$) is less than or equal to $d$.