Graph product

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In mathematics, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs G1 and G2 and produces a graph H with the following properties:

  • The vertex set of H is the Cartesian product V(G1) × V(G2), where V(G1) and V(G2) are the vertex sets of G1 and G2, respectively.
  • Two vertices (u1u2) and (v1v2) of H are connected by an edge if and only if the vertices u1, u2, v1, v2 satisfy conditions of a certain type (see below).

The following table shows the most common graph products, with ; denoting “is connected by an edge to”, and denoting non-connection. The operator symbols listed here are by no means standard, especially in older papers.

Name Condition for () ∼ (). Dimensions Example
Cartesian product
 =  and    )

   and  =  )

Tensor product
(Categorical product)
   and     Graph-tensor-product.svg
Lexicographical product
u1 ∼ v1
u1 = v1 and u2 ∼ v2 )
Strong product
(Normal product, AND product)
u1 = v1 and u2 ∼ v2 )
u1 ∼ v1 and u2 = v2 )
u1 ∼ v1 and u2 ∼ v2 )
Co-normal product
(disjunctive product, OR product)
u1 ∼ v1
u2 ∼ v2
Modular product

Rooted product see article Graph-rooted-product.svg
Zig-zag product see article see article see article
Replacement product
Homomorphic product[1][3]


In general, a graph product is determined by any condition for (u1u2) ∼ (v1v2) that can be expressed in terms of the statements u1 ∼ v1, u2 ∼ v2, u1 = v1, and u2 = v2.


Let be the complete graph on two vertices (i.e. a single edge). The product graphs , , and look exactly like the graph representing the operator. For example, is a four cycle (a square) and is the complete graph on four vertices. The notation for lexicographic product serves as a reminder that this product is not commutative.

See also[edit]


  1. ^ a b Roberson, David E.; Mancinska, Laura (2012). "Graph Homomorphisms for Quantum Players". arXiv:1212.1724Freely accessible [quant-ph]. 
  2. ^ Bačík, R.; Mahajan, S. (1995). "Semidefinite programming and its applications to NP problems". Computing and Combinatorics. Lecture Notes in Computer Science. 959. p. 566. doi:10.1007/BFb0030878. ISBN 3-540-60216-X. 
  3. ^ The hom-product of [2] is the graph complement of the homomorphic product of.[1]


  • Imrich, Wilfried; Klavžar, Sandi (2000). Product Graphs: Structure and Recognition. Wiley. ISBN 0-471-37039-8{{inconsistent citations}} .