We calculate the K-theory of the Cuntz-Krieger algebra ${ \cal O}_E$ 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 $\beta(E)$ is the first Betti number and $\gamma(E)$ is the valency number of the graph $E$. 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: $K_1({ \cal O}_E)= { \mathbb Z}^{ \beta(E)}$. These allow us to provide a counterexample to the fact, which holds for finite graphs, that $K_1({ \cal O}_E)$ is the torsion free part of $K_0({ \cal O}_E)$.
Categories: math.OA math.GR math.KT math.RA