The graph join 
of graphs 
and 
with disjoint sets 
and 
of vertices 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].
In particular, if 
is the singleton graph, then the graph
corona product 
is the graph join 
.
A complete k-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  |
