A note on the isomorphism conjectures for Leavitt path algebras

We relate two conjectures which have been raised for classification of Leavitt path algebras. For purely infinite simple unital Leavitt path algebras, it is conjectured that K-0 classifies them completely (Abrams et al., 2008, 2011 [3,4]). For arbitrary unital Leavitt path algebras, it is conjectured that K-0(gr) classifies them completely (Hazrat, in press [12]). We show that for two finite graphs with no sinks (which their associated Leavitt path algebras include the purely infinite simple-ones) if their K-0(gr)-groups of their Leavitt path algebras are isomorphic then their K-0-groups are isomorphic as well. We also provide a short proof of the fact that for a finite graph, its associated Leavitt path algebra is strongly graded if and. only if the graph has no sinks.