  In the past, it has been shown that the Leavitt path algebra $L(E)=L_K(E)$ of
a graph $E$ over a field $K$ is left and right noetherian if and only if the
graph $E$ is finite and no cycle of $E$ has an exit. If $Q(E)=Q_K(E)$ denotes
the regular algebra over $L(E),$ we prove that these conditions are further
equivalent with any of the following: $L(E)$ contains no infinite set of
orthogonal idempotents, $L(E)$ has finite uniform dimension, $L(E)$ is directly
finite, $Q(E)$ is directly finite, $Q(E)$ is unit-regular, and a few more
equivalences. In addition, if the involution on $K$ is positive definite, these
conditions are equivalent with the following: the involution $^\ast$ extends
from $L(E)$ to $Q(E),$ $Q(E)$ is $^\ast$-regular, $Q(E)$ is finite, $Q(E)$ is
the maximal (total or classical) symmetric ring of quotients of $L(E),$ every
finitely generated nonsingular $L(E)$-module is projective, and the matrix ring
$M_n(L(E))$ is strongly Baer for every $n$. It may not be surprising that a
noetherian Leavitt path algebra has these properties, but a more interesting
fact is that these properties hold only if a Leavitt path algebra is
noetherian.
  Using some of these equivalences, we give a specific description of the
inverse of the isomorphism $V(L(E))\rightarrow V(Q(E))$ of monoids of
equivalence classes of finitely generated projective modules for noetherian
Leavitt path algebras. We also prove that two noetherian Leavitt path algebras
are isomorphic as rings if and only if they are isomorphic as $^\ast$-algebras.
This answers in affirmative the Isomorphism Conjecture for the class of
noetherian Leavitt path algebras: if $L_{\mathbb C}(E)$ and $L_{\mathbb C}(F)$
are noetherian Leavitt path algebras, then $L_{\mathbb C}(E)\cong L_{\mathbb
C}(F)$ as rings implies $C^*(E)\cong C^*(F)$ as $^\ast$-algebras.
