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.

Note that some authors (e.g., Gallian 2007) adopt an alternate convention in which denotes the wheel graph on nodes.

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