In
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, a graph product is a
binary operation on
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
s. 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
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed ...
,
iff
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicon ...
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.
Overview table
The following table shows the most common graph products, with
denoting "is connected by an edge to", and
denoting non-connection. 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
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