Graphs with disjoint cycles, classification via the talented monoid

Abstract

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 T E has a certain Jordan-Hölder composition series. These are graphs whose associated Leavitt path algebras have finite Gelfand-Kirillov dimension. We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since T E is the positive cone of the graded Grothendieck group K gr 0 (L k (E)), we conclude that for graphs E and F , if, thus providing more evidence for the Graded Classification Conjecture for Leavitt path algebras.