two vertices (u,v) and (u' ,v' ) are adjacent in G □ Hif and only if either
u = u' and v is adjacent to v' in H, or
v = v' and u is adjacent to u' in G.
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.[1]
Examples
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.
Properties
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.[2] 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 and 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.[3]
A Cartesian product is
bipartite if and only if each of its factors is. More generally, the
chromatic number of the Cartesian product satisfies the equation
Cartesian product graphs can be recognized efficiently, in
linear time.[6]
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
,
where denotes the
Kronecker product of matrices and denotes the identity matrix.[7] The adjacency matrix of the Cartesian graph product is therefore the
Kronecker sum of the adjacency matrices of the factors.
Category theory
Viewing a graph as a
category whose objects are the vertices and whose morphisms are the paths in the graph, the cartesian product of graphs corresponds to the
funny tensor product of categories. The cartesian product of graphs is one of two graph products that turn the category of graphs and graph homomorphisms into a
symmetricclosed monoidal category (as opposed to merely symmetric monoidal), the other being the
tensor product of graphs.[8] The internal hom for the cartesian product of graphs has graph homomorphisms from to as vertices and "
unnatural transformations" between them as edges.[8]
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.