As defined in this work, a wheel graph 
of order 
, sometimes simply called an 
-wheel (Harary 1994, p. 46; Pemmaraju and Skiena 2003,
p. 248; Tutte 2005, p. 78), is a graph that contains a cycle
of order 
and for which every graph vertex in the cycle is
connected to one other graph vertex known as the
hub. 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 
, where 
is the singleton graph
and 
is the cycle
graph, making it a 
-cone graph. Wheel graphs of order 4 to 8 are illustrated
above in a number of embeddings.
Note that some authors (e.g., Gallian 2007) 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.
The wheel graph 
has graph dimension 2 for 
(and hence is unit-distance)
and dimension 3 otherwise (and hence 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 degree 
, and other nodes have degree 3. Wheel
graphs are 3-connected. 
,
where 
is the complete graph of 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)
|
