AbstractUsing 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$C^{\star }$-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.