The join 
of graphs 
and 
with disjoint point sets 
and 
and edge sets 
and 
is the graph union 
together with all the edges joining 
and 
(Harary 1994, p. 21). Graph joins are implemented in the Wolfram Language as GraphJoin[G1, G2].
A complete 
-partite graph 
is the graph join of empty graphs on 
, 
, ... nodes. A wheel graph is the join of a cycle graph and the singleton graph. Finally, a star graph is the join of an empty graph and the singleton graph (Skiena 1990, p. 132).
The following table gives examples of some graph joins. Here 
denotes an empty graph (i.e., the graph complement of the complete graph 
), 
a cycle graph, and 
the singleton graph.
| product | result |
 | complete  -partite graph  |
 | wheel graph  |
 | star graph  |
 |  -dipyramidal graph |
 | cone graph  |
 | fan graph |
 | windmill graph  |
