In graph theory, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs and and produces a graph with the following properties: The vertex set of is the Cartesian product , where and are the vertex sets of and , respectively. Two vertices and of are connected by an edge, iff a condition about in and in is fulfilled.
The graph products differ in what exactly this condition is. It is always about whether or not the vertices in 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 , and not as the formula would suggest.
Overview table The following table shows the most common graph products, with denoting "is connected by an edge to", and denoting non-adjacency. While does allow equality, 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 Number of edges Example with abbreviated as Cartesian product(box product) Tensor product(Kronecker product,categorical product) Strong product(Normal product, AND product) Lexicographical product or Co-normal product(disjunctive product,Graph Products Revisited: Tight Approximation Hardness of Induced Matching, Poset Dimension and More, Parinya Chalermsook, Bundit Laekhanukit, Danupon Nanongkai, 2012 OR product) or Modular product Rooted product see article Zig-zag product see article see article see article Replacement product Homomorphic product
In general, a graph product is determined by any condition for that can be expressed in terms of and .
Mnemonic
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. The resulting graph looks like substituting a copy of for every vertex of .
See also Graph operations
Notes
References