Using a result of Vdovina, we may associate to each complete connected bipartite graph $κ$ a $2$-dimensional square complex, which we call a tile complex, whose link at each vertex is $κ$. 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.