Rook Graph -- from Wolfram MathWorld

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.

Define an Latin square graph as a graph whose vertices are the elements of the Latin square and such that two vertices being are if they lie in the same row or column or contain the same symbol. These graphs correspond to the rook graph and the minimum vertex colorings of the rook graph give the distinct Latin squares.

rook graphs are distance-regular and geometric.

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.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.

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.

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.

Rook Graph

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