connected to each other through the other two vertices that use a combination of the same two ranks and files. As they show, except in the case , these properties uniquely characterize the rook's graph. That is, the rook's graphs are the only graphs with these numbers of vertices, edges, triangles per edge, and with a unique 4-cycle through each two non-adjacent vertices. When , these conditions may be abbreviated by stating that an rook's graph is a strongly regular graph with parameters . These parameters describe the number of vertices, the number of edges per vertex, the number of triangles per edge, and the number of shared neighbors for two non-adjacent vertices, respectively. Conversely, every strongly regular graph with these parameters must be an rook's graph, unless . When , there is another strongly regular graph, the Shrikhande graph, with the same parameters as the rook's graph. The Shrikhande graph obeys the same properties listed by Moon and Moser. It can be distinguished from the rook's graph in that the neighborhood of each vertex in the Shrikhande graph is connected to form a . In contrast, in the rook's graph, the neighborhood of each vertex forms two triangles, one for its rank and another for its file, without any edges from one part of the neighborhood to the other. Another way of distinguishing the rook's graph from the Shrikhande graph uses clique cover numbers: the rook's graph can be covered by four cliques (the four ranks or the four files of the chessboard) whereas six cliques are needed to cover the Shrikhande graph. Symmetry Rook's graphs are vertex-transitive, meaning that they have symmetries taking every vertex to every other vertex. This implies that every vertex has an equal number of edges: they are -regular. The rook's graphs are the only regular graphs formed from the moves of standard chess pieces in this way. When , the symmetries of the rook's graph are formed by independently permuting the rows and columns of the graph, so the automorphism group of the graph has elements. When , the graph has additional symmetries that swap the rows and columns, so the number of automorphisms is . Any two vertices in a rook's graph are either at distance one or two from each other, according to whether they are adjacent or nonadjacent respectively. Any two nonadjacent vertices may be transformed into any other two