View source for Cartesian product of graphs

{{short description|Operation in graph theory}}
[[File:Graph-Cartesian-product.svg|thumb|upright=1.35|A Cartesian product of two graphs]]

In [[graph theory]], the '''Cartesian product''' {{math|''G'' □ ''H''}}  of [[Graph (discrete mathematics)|graph]]s {{mvar|G}} and {{mvar|H}} is a graph such that:
* the [[Vertex (graph theory)|vertex]] set of {{math|''G'' □ ''H''}} is the [[Cartesian product]] {{math|''V''(''G'') × ''V''(''H'')}}; and
* two vertices {{math|(''u'',''v'')}} and {{math|(''u' '',''v' '')}} are adjacent in {{math|''G'' □ ''H''}} [[if and only if]] either 
** {{math|1=''u'' = ''u' ''}} and {{mvar|v}} is adjacent to {{mvar|v'}} in {{mvar|H}}, '''or'''
**  {{math|1=''v'' = ''v' ''}} and {{mvar|u}} is adjacent to {{mvar|u'}} in {{mvar|G}}.
The Cartesian product of graphs is sometimes called the '''box product''' of graphs [Harary 1969].

The operation is [[associative]], as the graphs  {{math|(''F'' □ ''G'') □ ''H''}} and {{math|''F'' □ (''G'' □ ''H'')}} are naturally [[Graph isomorphism|isomorphic]].
The operation is [[commutative]] as an operation on isomorphism [[equivalence class|classes]] of graphs, and more strongly the graphs {{math|''G'' □ ''H''}} and {{math|''H'' □ ''G''}} are [[natural isomorphism|naturally isomorphic]], but it is not commutative as an operation on [[Graph labeling|labeled graphs]]. 

The notation {{math|''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.{{sfnp|Hahn|Sabidussi|1997}}

==Examples==
* The Cartesian product of two edges is a [[cycle graph|cycle]] on four vertices: K<sub>2</sub>{{math|□}}K<sub>2</sub> = C<sub>4</sub>.
* The Cartesian product of K<sub>2</sub> and a [[path graph]] is a [[ladder graph]].
* The Cartesian product of two path graphs is a [[grid graph]].
* The Cartesian product of ''n'' edges is a hypercube: 
:: <math>(K_2)^{\square n} = Q_n.</math>
:Thus, the Cartesian product of two [[hypercube graph]]s is another hypercube: Q<sub>i</sub>{{math|□}}Q<sub>j</sub> = Q<sub>i+j</sub>. 
* The Cartesian product of two [[median graph]]s is another median graph.
* The graph of vertices and edges of an n-[[prism (geometry)|prism]] is the Cartesian product graph K<sub>2</sub>{{math|□}}C<sub>n</sub>.
* 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.<ref>{{harvtxt|Sabidussi|1960}}; {{harvtxt|Vizing|1963}}.</ref> However, {{harvtxt|Imrich|Klavžar|2000}} describe a disconnected graph that can be expressed in two different ways as a Cartesian product of prime graphs:
:<math>(K_1 + K_2 + K_2^2) \mathbin{\square} (K_1 + K_2^3) = (K_1 + K_2^2 + K_2^4) \mathbin{\square} (K_1 + K_2),</math>
where the plus sign denotes [[disjoint union]] and the superscripts denote exponentiation over Cartesian products. This is related to the identity that
:<math>
\begin{align}
(1 + x + x^2)(1 + x^3) &= (1 + x^2 + x^4)(1 + x) \\
                       &= 1 + x + x^2 + x^3 + x^4 + x^5 \\
                       &= (1 + x)(1 + x + x^2)(1 - x + x^2)
\end{align}
</math>
Both the factors <math>1+x^3</math> and <math>1+x^2+x^4</math> are not [[irreducible polynomial]]s, 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 domain|unique factorization]] property.

A Cartesian product is [[vertex-transitive graph|vertex transitive]] if and only if each of its factors is.<ref>{{harvtxt|Imrich|Klavžar|2000}}, Theorem&nbsp;4.19.</ref>

A Cartesian product is [[bipartite graph|bipartite]] if and only if each of its factors is. More generally, the [[chromatic number]] of the Cartesian product satisfies the equation 
:<math>\chi (G \mathbin{\square} H) = \max \{ \chi (G), \chi (H) \}.</math> {{sfnp|Sabidussi|1957}}
The [[Hedetniemi conjecture]] states a related equality for the [[tensor product of graphs]]. The independence number of a Cartesian product is not so easily calculated, but as {{harvtxt|Vizing|1963}} showed it satisfies the inequalities
:<math>\alpha (G) \alpha (H) + \min \{ |V(G)| -\alpha (G), |V(H)| - \alpha (H) \} \le \alpha (G \mathbin{\square} H) \le \min \{ \alpha (G) |V(H)|, \alpha (H) |V(G)| \}.</math>
The [[Vizing conjecture]] states that the [[domination number]] of a Cartesian product satisfies the inequality
:<math>\gamma (G \mathbin{\square} H) \ge \gamma (G) \gamma (H).</math>

The Cartesian product of [[unit distance graph]]s is another unit distance graph.{{sfnp|Horvat|Pisanski|2010}}

Cartesian product graphs can be recognized efficiently, in [[linear time]].<ref>{{harvtxt|Imrich|Peterin|2007}}. For earlier [[polynomial time]] algorithms see {{harvtxt|Feigenbaum|Hershberger|Schäffer|1985}} and {{harvtxt|Aurenhammer|Hagauer|Imrich|1992}}.</ref>

== Algebraic graph theory ==
[[Algebraic graph theory]] can be used to analyse the Cartesian graph product.
If the graph <math>G_1</math> has <math>n_1</math> vertices and the <math>n_1\times n_1</math> [[adjacency matrix]] <math>\mathbf A_1</math>, and the graph <math>G_2</math> has <math>n_2</math> vertices and the <math>n_2\times n_2</math> adjacency matrix <math>\mathbf A_2</math>, then the adjacency matrix of the Cartesian product of both graphs is given by
: <math>\mathbf A_{1\mathbin{\square} 2} = \mathbf A_1 \otimes \mathbf I_{n_2} + \mathbf I_{n_1} \otimes \mathbf A_2</math>,
where <math>\otimes</math> denotes the [[Kronecker product]] of matrices and <math>\mathbf I_n</math> denotes the <math>n\times n</math> [[identity matrix]].{{sfnp|Kaveh|Rahami|2005}} 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 (mathematics)|category]] whose objects are the vertices and whose morphisms are the paths in the graph, the cartesian product of graphs corresponds to the [https://ncatlab.org/nlab/show/funny+tensor+product 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 [[symmetric monoidal category|symmetric]] [[closed monoidal category]] (as opposed to merely symmetric monoidal), the other being the [[tensor product of graphs]].{{sfn|Weber|2013}}  The internal hom <math>[G, H]</math> for the cartesian product of graphs has graph homomorphisms from <math>G</math> to <math>H</math> as vertices and "[https://ncatlab.org/nlab/show/unnatural+transformation unnatural transformations]" between them as edges.{{sfn|Weber|2013}}

== History ==
According to {{harvtxt|Imrich|Klavžar|2000}}, Cartesian products of graphs were defined in 1912 by [[Alfred North Whitehead|Whitehead]] and [[Bertrand Russell|Russell]].  They were repeatedly rediscovered later, notably by {{harvs|first=Gert|last=Sabidussi|authorlink=Gert Sabidussi|year=1960|txt}}.

==Notes==
{{reflist|30em}}

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== External links ==
*{{mathworld | title = Graph Cartesian Product | urlname = GraphCartesianProduct}}

[[Category:Graph products]]