K-theory of locally finite graph C∗-algebras

In the case of an infinite graph with the finite Betti number we obtain the formula for the Grothendieck group K0(OE)=Zβ(E)+γ(E), where β(E) is the first Betti number and γ(E) is the valency number of the graph E. We note that in the infinite case the torsion part of K0, which is present in the case of a finite graph, vanishes. The Whitehead group depends only on the first Betti number: K1(OE)=Zβ(E). These allow us to provide a counterexample to the fact, which holds for finite graphs, that K1(OE) is the torsion free part of K0(OE).