Mapping graph homology to $K$-theory of Roe algebras

Given a graph $\mathrm{\Gamma }$, one may conside the set $X$ of its vertices as a metric space by assuming that all edges have length one. We consider two versions of homology theory of $\mathrm{\Gamma }$ and their $K$-theory counterparts – the $K$-theory of the (uniform) Roe algebra of the metric space $X$ of vertices of $\mathrm{\Gamma }$. We construct here a natural map from homology of $\mathrm{\Gamma }$ to the $K$-theory of the Roe algebra of $X$, and its uniform version. We show that, when $\mathrm{\Gamma }$ is the Cayley graph of $\mathbb{Z}$, the constructed maps are isomorphisms.