Graph product

In graph theory, 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 (a₁,a₂) and (b₁,b₂) 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 a_n, b_n in G_n 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.

Even for more standard definitions, it is not always consistent in the literature how to handle self-loops. The formulas below for the number of edges in a product also may fail when including self-loops. For example, the tensor product of a single vertex self-loop with itself is another single vertex self-loop with $E=1$, and not $E=2$ as the formula $E_{G\times H}=2E_{G}E_{H}$ would suggest.

Overview table

The following table shows the most common graph products, with $\sim$ denoting "is connected by an edge to", and $\not \sim$ denoting non-adjacency. While $\not \sim$ does allow equality, $\not \simeq$ means they must be distinct and non-adjacent. The operator symbols listed here are by no means standard, especially in older papers.

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

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

Mnemonic

Let $K_{2}$ be the complete graph on two vertices (i.e. a single edge). The product graphs $K_{2}\square K_{2}$, $K_{2}\times K_{2}$, and $K_{2}\boxtimes K_{2}$ look exactly like the graph representing the operator. For example, $K_{2}\square K_{2}$ is a four cycle (a square) and $K_{2}\boxtimes K_{2}$ is the complete graph on four vertices.

The $G_{1}[G_{2}]$ notation for lexicographic product serves as a reminder that this product is not commutative. The resulting graph looks like substituting a copy of $G_{2}$ for every vertex of $G_{1}$.

See also

• Graph operations

References

• Imrich, Wilfried; Klavžar, Sandi (2000). Product Graphs: Structure and Recognition. Wiley. ISBN 978-0-471-37039-0.