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  |
