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

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