Cartesian product of graphs
In graph theory, the Cartesian product "G""H" of graphs "G" and "H" is a graph such that The Cartesian product of graphs is sometimes called the box product of graphs [Harary 1969]. The operation is associative, as the graphs ("F'G")"H" and "F"("G'H") are naturally isomorphic. The operation is commutative as an operation on isomorphism classes of graphs, and more strongly the graphs "G'H" and "H'G" are naturally isomorphic, but it is not commutative as an operation on labeled graphs. The notation "G" × "H" has often been used for Cartesian products of graphs, but is now  more commonly used for another construction known as the tensor product of graphs. The square symbol is intended to be an intuitive and unambiguous notation for the Cartesian product, since it shows visually the four edges resulting from the Cartesian product of two edges. Properties. If a connected graph is a Cartesian product, it can be factorized uniquely as a product of prime factors, graphs that cannot themselves be decomposed as products of graphs. However, describe a disconnected graph that can be expressed in two different ways as a Cartesian product of prime graphs: where the plus sign denotes  disjoint union and the superscripts denote exponentiation over Cartesian products. A Cartesian product is vertex transitive if and only if each of its factors is. A Cartesian product is bipartite if and only if each of its factors is. More generally, the chromatic number of the Cartesian product satisfies the equation The Hedetniemi conjecture states a related equality for the tensor product of graphs. The independence number of a Cartesian product is not so easily calculated, but as showed it satisfies the inequalities The Vizing conjecture states that the domination number of a Cartesian product satisfies the inequality The Cartesian  product of unit distance graphs is another unit distance graph. Cartesian product graphs can be recognized efficiently, in linear time. Algebraic graph theory. Algebraic graph theory can be used to analyse the Cartesian graph product. If the graph formula_2 has formula_3 vertices and the formula_4 adjacency matrix formula_5, and the graph formula_6 has formula_7 vertices and the formula_8 adjacency matrix formula_9, then the adjacency matrix of the Cartesian product of both graphs is given by where formula_11 denotes the Kronecker product of matrices and formula_12 denotes the formula_13 identity matrix. The adjacency matrix of the Cartesian graph product is therefore  the Kronecker sum of the adjacency matrices of the factors. Category theory. Viewing a graph as a category whose objects are the vertices and whose morphisms are the paths in the graph, the cartesian product of graphs corresponds to the funny tensor product of categories. The cartesian product of graphs is one of two graph products that turn the category of graphs and graph homomorphisms into a symmetric closed monoidal category (as opposed to merely symmetric monoidal), the other being the tensor product of graphs. The internal hom formula_14 for the cartesian product of graphs has graph homomorphisms from formula_15  to formula_16 as vertices and "unnatural transformations" between them as edges. History. According to , Cartesian products of graphs were defined in 1912 by Whitehead and Russell. They were repeatedly rediscovered later, notably by .