# Graph product

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 (a1a2) and (b1b2) of H are connected by an edge, iff a condition about a1, b1 in G1 and a2, b2 in G2 is fulfilled.

The graph products differ in what exactly this condition is. It is always about whether or not the vertices an, bn in Gn are equal or connected by an edge.

The terminology and notation for specific graph products in the literature varies quite a lot; even if the following may be considered somewhat standard, readers are advised to check what definition a particular author uses for a graph product, especially in older texts.

## Overview table

The following table shows the most common graph products, with ${\displaystyle \sim }$ denoting “is connected by an edge to”, and ${\displaystyle \not \sim }$ denoting non-connection. The operator symbols listed here are by no means standard, especially in older papers.

Name Condition for ${\displaystyle (a_{1},a_{2})\sim (b_{1},b_{2})}$ Number of edges
${\displaystyle {\begin{array}{cc}v_{1}=\vert \mathrm {V} (G_{1})\vert &v_{2}=\vert \mathrm {V} (G_{2})\vert \\e_{1}=\vert \mathrm {E} (G_{1})\vert &e_{2}=\vert \mathrm {E} (G_{2})\vert \end{array}}}$
Example
with ${\displaystyle a_{n}~{\text{rel}}~b_{n}}$
abbreviated as ${\displaystyle {\text{rel}}_{n}}$
Cartesian product
${\displaystyle G_{1}\square G_{2}}$
${\displaystyle a_{1}=b_{1}~\land ~a_{2}\sim b_{2}}$
${\displaystyle \lor }$
${\displaystyle a_{1}\sim b_{1}~\land ~a_{2}=b_{2}}$
${\displaystyle =_{1}~\sim _{2}}$
${\displaystyle \lor }$
${\displaystyle \sim _{1}~=_{2}}$
${\displaystyle v_{1}~e_{2}~+~e_{1}~v_{2}}$
Tensor product
(Kronecker product,
categorical product)
${\displaystyle G_{1}\times G_{2}}$
${\displaystyle a_{1}\sim b_{1}~\land ~a_{2}\sim b_{2}}$ ${\displaystyle \sim _{1}~\sim _{2}}$ ${\displaystyle 2~e_{1}~e_{2}}$
Lexicographical product
${\displaystyle G_{1}\cdot G_{2}}$ or ${\displaystyle G_{1}[G_{2}]}$
${\displaystyle a_{1}\sim b_{1}}$
${\displaystyle \lor }$
${\displaystyle a_{1}=b_{1}~\land ~a_{2}\sim b_{2}}$
${\displaystyle \sim _{1}}$
${\displaystyle \lor }$
${\displaystyle =_{1}~\sim _{2}}$
${\displaystyle v_{1}~e_{2}~+~e_{1}~v_{2}^{2}}$
Strong product
(Normal product,
AND product)
${\displaystyle G_{1}\boxtimes G_{2}}$
${\displaystyle a_{1}=b_{1}~\land ~a_{2}\sim b_{2}}$
${\displaystyle \lor }$
${\displaystyle a_{1}\sim b_{1}~\land ~a_{2}=b_{2}}$
${\displaystyle \lor }$
${\displaystyle a_{1}\sim b_{1}~\land ~a_{2}\sim b_{2}}$
${\displaystyle =_{1}~\sim _{2}}$
${\displaystyle \lor }$
${\displaystyle \sim _{1}~=_{2}}$
${\displaystyle \lor }$
${\displaystyle \sim _{1}~\sim _{2}}$
${\displaystyle v_{1}~e_{2}~+~e_{1}~v_{2}~+~2~e_{1}~e_{2}}$
Co-normal product
(disjunctive product, OR product)
${\displaystyle G_{1}*G_{2}}$
${\displaystyle a_{1}\sim b_{1}}$
${\displaystyle \lor }$
${\displaystyle a_{2}\sim b_{2}}$
${\displaystyle \sim _{1}}$
${\displaystyle \lor }$
${\displaystyle \sim _{2}}$
${\displaystyle v_{1}^{2}~e_{2}~+~e_{1}~v_{2}^{2}~-~2~e_{1}~e_{2}}$
Modular product ${\displaystyle a_{1}\sim b_{1}~\land ~a_{2}\sim b_{2}}$
${\displaystyle \lor }$
${\displaystyle a_{1}\not \sim b_{1}~\land ~a_{2}\not \sim b_{2}}$
${\displaystyle \sim _{1}~\sim _{2}}$
${\displaystyle \lor }$
${\displaystyle \not \sim _{1}~\not \sim _{2}}$
Rooted product see article ${\displaystyle v_{1}~e_{2}~+~e_{1}}$
Zig-zag product see article see article see article
Replacement product
Homomorphic product[1][3]
${\displaystyle G_{1}\ltimes G_{2}}$
${\displaystyle a_{1}=b_{1}}$
${\displaystyle \lor }$
${\displaystyle a_{1}\sim b_{1}~\land ~a_{2}\not \sim b_{2}}$
${\displaystyle =_{1}}$
${\displaystyle \lor }$
${\displaystyle \sim _{1}~\not \sim _{2}}$

In general, a graph product is determined by any condition for ${\displaystyle (a_{1},a_{2})\sim (b_{1},b_{2})}$ that can be expressed in terms of ${\displaystyle a_{n}=b_{n}}$ and ${\displaystyle a_{n}\sim b_{n}}$.

## Mnemonic

Let ${\displaystyle K_{2}}$ be the complete graph on two vertices (i.e. a single edge). The product graphs ${\displaystyle K_{2}\square K_{2}}$, ${\displaystyle K_{2}\times K_{2}}$, and ${\displaystyle K_{2}\boxtimes K_{2}}$ look exactly like the graph representing the operator. For example, ${\displaystyle K_{2}\square K_{2}}$ is a four cycle (a square) and ${\displaystyle K_{2}\boxtimes K_{2}}$ is the complete graph on four vertices. The ${\displaystyle G_{1}[G_{2}]}$ notation for lexicographic product serves as a reminder that this product is not commutative.