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)
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(2)
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(3)
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where . These correspond
to recurrence equations (together with for the rank
polynomial) of
(4)
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(5)
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(6)
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(7)
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