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: V(G1) × V(G2) , where V(G1) and V(G2) are the vertex sets of G1 and G2 , respectively. (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 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 E=1, and not E=2 as the formula 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. In general, a graph product is determined by any condition for that can be expressed in terms of a_n = b_n and.

Mnemonic

Let K_2 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 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.