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Cone Graph

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An m-gonal n-cone graph, also called the n-point suspension of C_m or generalized wheel graph (Buckley and Harary 1988), is defined by the graph join C_m+K^__n, where C_m is a cycle graph and K^__n is an empty graph (Gallian 2025, p. 15), with integers m>=3 and n>=1.

Special cases are summarized in the following table.

(m,n)graph
(3,1)tetrahedral graph K_4
(4,2)octahedral graph K_(2,2,2)
(4,n)complete tripartite graph K_(2,2,n)
(m,1)(m+1)-wheel graph
(m,2)m-dipyramidal graph

Buckley and Harary (1988) consider the graph dimension of the cone graph.

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

ConeGraph

Embeddings of a number of small cone graphs are illustrated above.

See also

Complete Tripartite Graph, Dipyramidal Graph, Graph Join, k-Tree, Wheel Graph

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References

Buckley, F. and Harary, F. "On the Euclidean Dimension of a Wheel." Graphs and Combin. 4, 23-30, 1988. https://doi.org/10.1007/BF01864150.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. Cone Graphs. Tetrahedron K_4, Wheel W_5, Wheel W_6, Wheel W_7, Wheel W_8, and others.

Referenced on Wolfram|Alpha

Cone Graph

Cite this as:

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

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