Graph Join

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 can be computed using GraphJoin[G1, G2] in the Wolfram Language package Combinatorica` .

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

REFERENCES

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.

Skiena, S. "Joins of Graphs." §4.1.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 131-132, 1990.