Graphs with disjoint cycles, classification via the talented monoid

We characterise directed graphs consisting of disjoint cycles via their talented monoids. We show that a graph E consists of disjoint cycles precisely when its talented monoid has a certain Jordan-Hölder composition series. These are graphs E whose associated Leavitt path algebras have finite Gelfand-Kirillov dimension (GKdim). We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since is the positive cone of the graded Grothendieck group , we conclude that for graphs E and F, if then , thus providing more evidence for the Graded Classification Conjecture for Leavitt path algebras.