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A wheel graph of graph order , where is an integer, is the graph obtained by adjoining a hub to the cycle graph and connecting it to every graph vertex of the cycle graph. As defined in this work, it is sometimes simply called an -wheel (Harary 1994, p. 46; Pemmaraju and Skiena 2003, p. 248; Tutte 2005, p. 78). The edges of a wheel which include the hub are called spokes (Skiena 1990, p. 146). The wheel can be defined as the graph join , or equivalently as the graph corona product , where is the singleton graph and is the cycle graph, making it an -cone graph. Wheel graphs of graph order 4 to 8 are illustrated above in a number of graph embeddings.

This graph-theoretic use of "wheel" is unrelated to the algebraic structure known as a wheel algebra.

Note that some authors (e.g., Gallian 2026) adopt an alternate convention in which denotes the wheel graph on nodes.

The tetrahedral graph (i.e., ) is isomorphic to , and is isomorphic to the complete tripartite graph . In general, the -wheel graph is the skeleton of an -pyramid.

The wheel graph is isomorphic to the Jahangir graph .

is one of the two graphs obtained by removing two edges from the pentatope graph , the other being the house X graph.

is a quasi-regular graph.

Wheel graphs are graceful (Frucht 1979), self-dual, pancyclic, and dominating unique (Akbari and Oboudi 2014, p. 358).

The wheel graph has graph dimension 2 for (and hence is unit-distance) and graph dimension 3 otherwise (and hence is not unit-distance) (Erdős et al. 1965, Buckley and Harary 1988).

Wheel graphs can be constructed in the Wolfram Language using WheelGraph[n]. Precomputed properties of a number of wheel graphs are available via GraphData[{"Wheel", n}].

The number of graph cycles in the wheel graph is given by , or 7, 13, 21, 31, 43, 57, ... (OEIS A002061) for , 5, ....

In a wheel graph, the hub has vertex degree , and other nodes have vertex degree 3. Wheel graphs are 3-connected. , where is the complete graph of graph order four. The chromatic number of is

The wheel graph has chromatic polynomial

See also

Complete Graph, Cone Graph, Dipyramidal Graph, Gear Graph, Helm Graph, Hub, Jahangir Graph, Ladder Graph, Pyramid, Spoke Graph, Tutte's Wheel Theorem, Web Graph, Wheel Algebra, Wheel Complement Graph

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References

Akbari, S. and Oboudi, M. R. "Cycles Are Determined by Their Domination Polynomials." Ars Combin. 116, 353-358, 2014.Brandstädt, A.; Le, V. B.; and Spinrad, J. P. Graph Classes: A Survey. Philadelphia, PA: SIAM, p. 19, 1999.Buckley, F. and Harary, F. "On the Euclidean Dimension of a Wheel." Graphs and Combin. 4, 23-30, 1988.Erdős, P.; Harary, F.; and Tutte, W. T. "On the Dimension of a Graph." Mathematika 12, 118-122, 1965.Frucht, R. "Graceful Numbering of Wheels and Related Graphs." Ann. New York Acad. Sci. 319, 219-229, 1979.Gallian, J. A. §2.2 in "A Dynamic Survey of Graph Labeling." Elec. J. Combin., Dynamic Survey DS6, Nov. 6, 2026. https://doi.org/10.37236/27.Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 46 and 167-168, 1994.House of Graphs. Wheel Graphs. Wheel W_5, Wheel W_6, Wheel W_7, Wheel W_8, Wheel W_9, and others.Pemmaraju, S. and Skiena, S. "Cycles, Stars, and Wheels." §6.2.4 in Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica. Cambridge, England: Cambridge University Press, pp. 248-249, 2003.Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 148, 1986.Skiena, S. "Cycles, Stars, and Wheels." §4.2.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 91 and 144-147, 1990.Sloane, N. J. A. Sequence A002061/M2638 in "The On-Line Encyclopedia of Integer Sequences."Tutte, W. T. Graph Theory. Cambridge, England: Cambridge University Press, 2005.

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Wheel Graph

Cite this as:

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

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