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 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.

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

The following table shows the most common graph products, with

denoting "is connected by an edge to", and

denoting non-adjacency.

means they must be distinct and non-adjacent.

The operator symbols listed here are by no means standard, especially in older papers.

In general, a graph product is determined by any condition for

that can be expressed in terms of

be the complete graph on two vertices (i.e. a single edge).

The product graphs

look exactly like the graph representing the operator.

is a four cycle (a square) and

is the complete graph on four vertices.

notation for lexicographic product serves as a reminder that this product is not commutative.

The resulting graph looks like substituting a copy of