Rook Graph

The  rook graph (confusingly called the  grid by Brouwer et al. 1989, p. 440) and also sometimes known as a lattice graph (e.g., Brouwer) is the graph Cartesian product  of complete graphs, which is equivalent to the line graph  of the complete bipartite graph . This is the definition adopted for example by Brualdi and Ryser (1991, p. 153), although restricted to the case . This definition corresponds to the connectivity graph of a rook chess piece (which can move any number of spaces in a straight line-either horizontally or vertically, but not diagonally) on an  chessboard.

The graph  has  vertices and  edges. It is regular of degree , has diameter 3, girth 3 (for ), and chromatic number . It is also perfect (since it is the line graph of a bipartite graph) and vertex-transitive.

The  rook graph  is also isomorphic to the Latin square graph. The vertices of such a graph are defined as the  elements of a Latin square of order , with two vertices being adjacent if they lie in the same row or column or contain the same symbol. It turns out that all Latin squares of order  produce the same rook graph.

Precomputed properties of rook graphs are implemented in the Wolfram Language as GraphData["Rook", m, n].

A rook graph  is a circulant graph iff  (i.e.,  is relatively prime to ). In that case, the rook graph is isomorphic to .

Special cases are summarized in the following table.

	isomorphic to
	square graph
	prism graph
	circulant graph
	graph complement of the -crown graph
	generalized quadrangle
	circulant graph
	25-cyclotomic graph

The following table summarized the bipartite double graphs of the rook graph  for small .

	bipartite double graph of
2
3	tesseract graph
4	prism graph
5	Kummer graph
5	Haar graph

A closed formula for the number of 7-cycles of  is given by

(Perepechko and Voropaev).

The  rook graph has domination number .

Aubert and Schneider (1982) showed that rook graphs admit Hamiltonian decomposition, meaning they are class 1 when they have even vertex count and class 2 when they have odd vertex count (because they are odd regular).

REFERENCES

Aubert, J. and Schneider, B. "Décomposition de la somme cartésienne d'un cycle et de l'union de deux cycles hamiltoniens en cycles hamiltoniens." Disc. Math. 38, 7-16, 1982.

Brouwer, A. E. "Lattice Graphs." http://www.win.tue.nl/~aeb/drg/graphs/Hamming.html.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.

Brouwer, A. E. and van Lint, J. H. "Strongly Regular Graphs and Partial Geometries." In Enumeration and Design: Papers from the conference on combinatorics held at the University of Waterloo, Waterloo, Ont., June 14-July 2, 1982 (Ed. D. M. Jackson and S. A. Vanstone). Toronto, Canada: Academic Press, pp. 85-122, 1984.

Brualdi, R. and Ryser, H. J. §6.2.4 in Combinatorial Matrix Theory. New York: Cambridge University Press, p. 152, 1991.

Godsil, C. and Royle, G. "Latin Square Graphs." §10.4 Algebraic Graph Theory. New York: Springer-Verlag, pp. 226-230, 2001.

Karavaev, A. M. "FlowProblem: Statistics of Simple Cycles." http://flowproblem.ru/paths/statistics-of-simple-cycles.Perepechko, S. N. and Voropaev, A. N. "The Number of Fixed Length Cycles in an Undirected Graph. Explicit Formulae in Case of Small Lengths."van Dam, E. R. and Haemers, W. H. "Which Graphs Are Determined by Their Spectrum?" Lin. Algebra Appl. 373, 139-162, 2003.