In graph theory, the tensor product G × H of graphs G and H is a graph such that

The tensor product is also called the direct product, Kronecker product, categorical product, cardinal product, relational product, weak direct product, or conjunction. As an operation on binary relations, the tensor product was introduced by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica (1912). It is also equivalent to the Kronecker product of the adjacency matrices of the graphs.

The notation G × H is also (and formerly normally was) used to represent another construction known as the Cartesian product of graphs, but nowadays more commonly refers to the tensor product. The cross symbol shows visually the two edges resulting from the tensor product of two edges. This product should not be confused with the strong product of graphs.

The tensor product is the category-theoretic product in the category of graphs and graph homomorphisms. That is, a homomorphism to G × H corresponds to a pair of homomorphisms to G and to H. In particular, a graph I admits a homomorphism into G × H if and only if it admits a homomorphism into G and into H.

To see that, in one direction, observe that a pair of homomorphisms f_G : I → G and f_H : I → H yields a homomorphism

In the other direction, a homomorphism f : I → G × H can be composed with the projections homomorphisms

to yield homomorphisms to G and to H.

The adjacency matrix of G × H is the Kronecker (tensor) product of the adjacency matrices of G and H.