 TOPICS  # Helm Graph The helm graph is the graph obtained from an -wheel graph by adjoining a pendant edge at each node of the cycle.

Helm graphs are graceful (Gallian 2018), with the odd case of established by Koh et al. 1980 and the even case by Ayel and Favaron (1984). The helm graph is perfect only for and even .

Precomputed properties of helm graphs are available in the Wolfram Language using GraphData[ "Helm", n, k  ].

The -Helm graph has chromatic polynomial, independence polynomial, and matching polynomial given by   (1)   (2)   (3)

where . These correspond to recurrence equations (together with for the rank polynomial) of   (4)   (5)   (6)   (7)

Crossed Prism Graph, Cycle Graph, Flower Graph, Möbius Ladder, Prism Graph, Web Graph, Wheel Graph

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## References

Ayel, J. and Favaron, O. "Helms Are Graceful. In Progress in Graph Theory (Waterloo, Ont., 1982). Toronto: Academic Press, pp. 89-92, 1984.Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Koh, K. M.; Rogers, D. G.; and Yap, K. Y. "Graceful Graphs: Some Further Results and Problems." Congr. Numer. 29, 559-571, 1980.Seoud, M. Z. and Youssef, M. A. "Harmonious Labelling of Helms and Related Graphs." Unpublished work. Jan. 2017. http://dx.doi.org/10.13140/RG.2.2.11041.61282.

Helm Graph

## Cite this as:

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