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.

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, 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. This is related to the identity that

Both the factors ${\displaystyle 1+x^{3}}$ and ${\displaystyle 1+x^{2}+x^{4}}$ 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.