 Square rook's graphs are connected-homogeneous, meaning that every isomorphism between two connected induced subgraphs can be extended to an automorphism of the whole graph.A rook's graph can also be viewed as the line graph of a complete bipartite graph — that is, it has one vertex for each edge of , and two vertices of the rook's graph are adjacent if and only if the corresponding edges of the complete bipartite graph share a common endpoint.In this view, an edge in the complete bipartite graph from the th vertex on one side of the bipartition to the th vertex on the other side corresponds to a chessboard square with coordinates .