The helm graph 
, for 
,
is the graph obtained from the wheel
graph 
on 
vertices
by adjoining a pendant edge at each node
of the rim cycle graph 
.
Helm graphs are graceful (Gallian 2026), 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}].
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/Inline13.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/Inline16.svg) |
(2)
|
 |  |  |
(3)
|
where 
. These correspond to recurrence
equations (together with those for the rank polynomial)
of
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
