Cartesian product of graphs

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 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.

[2] However, Imrich & Klavžar (2000) 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.

are not irreducible polynomials, but their factors include negative coefficients and thus the corresponding graphs cannot be decomposed.

In this sense, the failure of unique factorization on (possibly disconnected) graphs is akin to the statement that polynomials with nonnegative integer coefficients is a semiring that fails the unique factorization property.

A Cartesian product is vertex transitive if and only if each of its factors is.

[3] 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 Vizing (1963) 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.

[5] Cartesian product graphs can be recognized efficiently, in linear time.

, then the adjacency matrix of the Cartesian product of both graphs is given by where

denotes the Kronecker product of matrices and

[7] The adjacency matrix of the Cartesian graph product is therefore the Kronecker sum of the adjacency matrices of the factors.

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.

[8] According to Imrich & Klavžar (2000), Cartesian products of graphs were defined in 1912 by Whitehead and Russell.

They were repeatedly rediscovered later, notably by Gert Sabidussi (1960).