Helm Graph
المؤلف:
Ayel, J. and Favaron, O.
المصدر:
"Helms Are Graceful. In Progress in Graph Theory (Waterloo, Ont., 1982).
الجزء والصفحة:
...
20-3-2022
2016
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[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/HelmGraph/Inline7.svg" style="height:22px; width:6px" />"Helm", ![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/HelmGraph/Inline8.svg" style="height:22px; width:6px" />n, k![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/HelmGraph/Inline9.svg" style="height:22px; width:6px" />![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/HelmGraph/Inline10.svg" style="height:22px; width:6px" />].
The 
-Helm graph has chromatic polynomial, independence polynomial, and matching polynomial given by
where 
. These correspond to recurrence equations (together with for the rank polynomial) of
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.
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