  In this paper we completely describe graphically Leavitt path algebras with
bounded index of nilpotence. We show that the Leavitt path algebra $L_{K}(E)$
has index of nilpotence at most $n$ if and only if no cycle in the graph $E$
has an exit and there is a fixed positive integer $n$ such that the number of
distinct paths that end at any given vertex $v$ (including $v$, but not
including the entire cycle $c$ in case $v$ lies on $c$) is less than or equal
to $n$. Interestingly, the Leavitt path algebras having bounded index of
nilpotence turn out to be precisely those that satisfy a polynomial identity.
Furthermore, Leavitt path algebras with bounded index of nilpotence are shown
to be directly-finite and to be $\mathbb{Z}$-graded $\Sigma$-$V$ rings. As an
application of our results, we answer an open question raised in \cite{JST}
whether an exchange $\Sigma$-$V$ ring has bounded index of nilpotence.
