K-theory of locally finite graph $C^{\ast }$-algebras

We calculate the K-theory of the Cuntz-Krieger algebra ${\mathcal{O}}_E$ associated with an infinite, locally finite graph, via the Bass-Hashimoto operator. The formulae we get express the Grothendieck group and the Whitehead group in purely graph theoretic terms. We consider the category of finite (black-and-white, bi-directed) subgraphs with certain graph homomorphisms and construct a continuous functor to abelian groups. In this category $K_0$ is an inductive limit of $K$-groups of finite graphs, which were calculated in \cite{MM}. In the case of an infinite graph with the finite Betti number we obtain the formula for the Grothendieck group $K_0({\cal O}_E)= {\mathbb Z}^{\beta(E)+\gamma(E)},,$where$\beta(E)$isthefirstBettinumberand$\gamma(E)$isthevalencynumberofthegraph$E$. We note, that in the infinite case the torsion part of $K_0$, which is present in the case of a finite graph, vanishes. The Whitehead group depends only on the first Betti number: $K_1({\mathcal{O}}_E)={\mathbb{Z}}^{\beta (E)}$. These allow us to provide a counterexample to the fact, which holds for finite graphs, that $K_1({\cal O}_E)$isthetorsionfreepartof$K_0({\cal O}_E)$.