Tensor product of graphs

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

the vertex set of $G \times H$ is the Cartesian product $V (G) \times V (H)$ ; 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.

The notation $G \times H$ was formerly (and occasionally still is) used to represent the Cartesian product of graphs, which nowadays is usually written $G □ H$ . The cross symbol in $G \times H$ depicts the two edges resulting from the tensor product of two edges, whereas $G □ H$ has four resulting edges. This product should not be confused with the strong product of graphs.