A wheel graph with n vertices can also be defined as the 1-skeleton of an (n – 1)-gonal pyramid.

Some authors[1] write Wn to denote a wheel graph with n vertices (n ≥ 4); other authors[2] instead use Wn to denote a wheel graph with n + 1 vertices (n ≥ 3), which is formed by connecting a single vertex to all vertices of a cycle of length n. The former notation is used in the rest of this article and in the table on the right.

For odd values of n, Wn is a perfect graph with chromatic number 3: the vertices of the cycle can be given two colors, and the center vertex given a third color.

The k-wheel matroid is the graphic matroid of a wheel Wk+1, while the k-whirl matroid is derived from the k-wheel by considering the outer cycle of the wheel, as well as all of its spanning trees, to be independent.

The wheel W6 supplied a counterexample to a conjecture of Paul Erdős on Ramsey theory: he had conjectured that the complete graph has the smallest Ramsey number among all graphs with the same chromatic number, but Faudree and McKay (1993) showed W6 has Ramsey number 17 while the complete graph with the same chromatic number, K4, has Ramsey number 18.