THE K-THEORY OF THE C*-ALGEBRAS OF 2-RANK GRAPHS ASSOCIATED TO COMPLETE BIPARTITE GRAPHS

Using a result of Vdovina, we may associate to each complete connected bipartite graph kappa a two-dimensional square complex, which we call a tile complex, whose link at each vertex is kappa. We regard the tile complex in two different ways, each having a different structure as a 2-rank graph. To each 2-rank graph is associated a universal C*-algebra, for which we compute the K-theory, thus providing a new infinite collection of 2-rank graph algebras with explicit K-groups. We determine the homology of the tile complexes and give generalisations of the procedures to complexes and systems consisting of polygons with a higher number of sides.