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Windmill Graph

WindmillGraph

The (m,n)-windmill graph, denoted K_n^((m)) by Gallian (2011, p. 16), is the graph obtained by taking m copies of the complete graph K_n with a vertex in common. The (m,n)-windmill graph is therefore isomorphic to the graph join mK_(n-1)+K_1.

The (2,n)-windmill graph is isomorphic to the vertex contraction K_n·K_n and the (m,3)-windmill graph is isomorphic to the (m,3)-Dutch windmill graph.

Special cases are summarized in the following table.

(m,n)graph
(2,3)butterfly graph
(2,4)3-double cone graph
(2,n)K_n·K_n
(m,3)(m,3)-Dutch windmill graph

Windmill graphs are geodetic.

Gallian (2018) summarizes known results about the gracefulness of windmill graphs.

Precomputed properties of windmill graphs are implemented in the Wolfram Language as GraphData[{"Windmill", {m, n}}].

See also

Double Cone Graph, Dutch Windmill Graph, Graph Join, Windmill

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References

Benson, M. and Lee, S. M. "On Cordialness of Regular Windmill Graphs." Congr. Numer. 68, 45-58, 1989.Bermond, J. C. "Graceful Graphs, Radio Antennae and French Windmills." Graph Theory and Combinatorics. London, England: Pitman, pp. 18-37, 1979.Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin., Dynamic Survey DS6, Oct. 30, 2025. https://doi.org/10.37236/27.House of Graphs. Windmill Graphs. K1 + 2K3, K1 + 2K4, and 3K3 + K1.Koh, K. M.; Rogers, D. G.; Teo, H. K.; and Yap, K. Y. "Graceful Graphs: Some Further Results and Problems." Congr. Numer. 29, 559-571, 1980.

Referenced on Wolfram|Alpha

Windmill Graph

Cite this as:

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

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