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 an intuitive and unambiguous notation for the Cartesian product, since it shows visually the four edges resulting from the Cartesian product of two edges.
The Cartesian product of two edges is a cycle on four vertices: K2 K2 = C4.
The Cartesian product of two path graphs is a grid graph.
The Cartesian product of n edges is a hypercube:
Thus, the Cartesian product of two hypercube graphs is another hypercube: Qi Qj = Qi+j.
The Cartesian product of two median graphs is another median graph.
The graph of vertices and edges of an n-prism is the Cartesian product graph K2 Cn.
The rook's graph is the Cartesian product of two complete graphs.
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:
Cartesian product graphs can be recognized efficiently, in linear time.
Algebraic graph theory
Algebraic graph theory can be used to analyse the Cartesian graph product. If the graph has vertices and the adjacency matrix , and the graph has vertices and the adjacency matrix , then the adjacency matrix of the Cartesian product of both graphs is given by
Kaveh, A.; Rahami, H. (2005), "A unified method for eigendecomposition of graph products", Communications in Numerical Methods in Engineering with Biomedical Applications, 21 (7): 377–388, doi:10.1002/cnm.753, MR2151527.