The helm graph 
is the graph obtained from an 
-wheel graph by adjoining a
pendant edge at each node
of the cycle graph.
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
 |  | ![z[(1-z)^n(z-2)+(z-2)^n(z-1)^n]](/images/equations/HelmGraph/Inline14.svg) |
(1)
|
 |  | ![2^(-n)[2^nx(+x+1)^n+(x-sqrt((x+1)(5x+1))+1)^n+(x+sqrt((x+1)(5x+1))+1)^n]](/images/equations/HelmGraph/Inline17.svg) |
(2)
|
 |  |  |
(3)
|
where 
. These correspond
to recurrence equations (together with those
for the rank polynomial) of
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
