  There is a tight relation between the geometry of a directed graph and the
algebraic structure of a Leavitt path algebra associated to it. In this note,
we show a similar connection between the geometry of the graph and the
structure of a certain monoid associated to it. This monoid is isomorphic to
the positive cone of the graded K0-group of the Leavitt path algebra which is
naturally equipped with a Z-action. As an example, we show that a graph has a
cycle without an exit if and only if the monoid has a periodic element.
Consequently a graph has Condition (L) if and only if the group Z acts freely
on the monoid. We go on to show that the algebraic structure of Leavitt path
algebras (such as simplicity, purely infinite simplicity, or the lattice of
ideals) can be described completely via this monoid. Therefore an isomorphism
between the monoids (or graded K0's) of two Leavitt path algebras implies that
the algebras have similar algebraic structures. These all confirm that the
graded Grothendieck group could be a sought-after complete invariant for the
classification of Leavitt path algebras.
