Tensor product of graphs

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

• the vertex set of G × H is the Cartesian product V(G) × V(H); and
• vertices (g,h) and (g',h' ) are adjacent in G × H if and only if
  • g is adjacent to g' in G, and
  • h is adjacent to h' in H.

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.^[1]

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.^[2] This product should not be confused with the strong product of graphs.