A wheel graph 
of graph order 
, where 
is an integer, is the graph obtained by adjoining a hub
to the cycle graph 
and connecting it to every graph
vertex of the cycle graph. As defined in this
work, it is sometimes simply called an 
-wheel (Harary 1994, p. 46; Pemmaraju and Skiena 2003,
p. 248; Tutte 2005, p. 78). The edges of
a wheel which include the hub are called spokes (Skiena 1990,
p. 146). The wheel 
can be defined as the graph join 
, or equivalently as the graph
corona product 
,
where 
is the singleton
graph and 
is the cycle graph, making it a 
-cone graph. Wheel graphs
of graph order 4 to 8 are illustrated above in a number
of graph embeddings.
This graph-theoretic use of "wheel" is unrelated to the algebraic structure known as a wheel algebra.
Note that some authors (e.g., Gallian 2025, p. 15) adopt an alternate convention in which 
denotes the wheel graph on 
nodes.
The tetrahedral graph (i.e., 
) is isomorphic to 
, and 
is isomorphic to the
complete tripartite graph 
. In general, the 
-wheel graph is the skeleton of
an 
-pyramid.
The wheel graph 
is isomorphic to the Jahangir
graph 
.
is one of the two graphs
obtained by removing two edges from the pentatope
graph 
,
the other being the house X graph.
is a quasi-regular
graph.
Wheel graphs are graceful (Frucht 1979), self-dual, pancyclic, and dominating unique (Akbari and Oboudi 2014, p. 358).
The wheel graph 
has graph dimension 2 for 
(and hence is unit-distance)
and graph dimension 3 otherwise (and hence is
not unit-distance) (Erdős et al. 1965,
Buckley and Harary 1988).
Wheel graphs can be constructed in the Wolfram Language using WheelGraph[n].
Precomputed properties of a number of wheel graphs are available via GraphData[
"Wheel", n
].
The number of graph cycles in the wheel graph 
is given by 
, or 7, 13, 21, 31, 43, 57, ... (OEIS A002061)
for 
, 5, ....
In a wheel graph, the hub has vertex degree 
,
and other nodes have vertex
degree 3. Wheel graphs are 3-connected.
, where 
is the complete graph
of graph order four. The chromatic
number of 
is
 |
(1)
|
The wheel graph 
has chromatic polynomial
![pi(x)=x[(x-2)^(n-1)-(-1)^n(x-2)].](/images/equations/WheelGraph/NumberedEquation2.svg) |
(2)
|
