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 .
The operation is associative, as the graphs H and F 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 an intuitive and unambiguous notation for the Cartesian product, since it shows visually the four edges resulting from the Cartesian product of two edges.

Examples

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

can be used to analyse the Cartesian graph product.
If the graph has vertices and the adjacency matrix, and the graph has vertices and the adjacency matrix, then the adjacency matrix of the Cartesian product of both graphs is given by
where denotes the Kronecker product of matrices and denotes the 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 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, the other being the tensor product of graphs. The internal hom for the cartesian product of graphs has graph homomorphisms from to as vertices and "" 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.