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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.

See also

Cone Graph, Dipyramidal Graph, Fan Graph, Graph Cartesian Product, Graph Sum, Graph Union, Simonovits Product Conjecture, Star Graph, Wheel Graph, Windmill Graph

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References

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 21 and 167-168, 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.

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Graph Join

Cite this as:

Weisstein, Eric W. "Graph Join." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GraphJoin.html

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