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 |