Gear Graph

The gear graph, also sometimes known as a bipartite wheel graph (Brandstädt et al. 1987), is a wheel graph with a graph vertex added between each pair of adjacent vertices of the outer graph cycle (Gallian 2025). The gear graph has vertices and edges.

The gear graphs are a special case of the Jahangir graph.

Gear graphs are unit-distance and matchstick graphs, as illustrated in the graph drawings shown above.

Derived unit-distance graphs are produced by taking the vertex sets from the matchstick graph drawings and connecting all pairs of vertices separated by a unit distance for , 6, 12, and 18, illustrated above, with the case corresponding to the wheel graph .

Ma and Feng (1984) proved that all gear graphs are graceful, and Liu (1996) showed that if two or more vertices are inserted between every pair of vertices of the outer graph cycle of the wheel graph, the resulting graph is also graceful (Gallian 2025).

For , the simplex graph of the cycle graph is the gear graph .

Precomputed properties of gear graphs are given in the Wolfram Language by GraphData["Gear", n].

The gear graph has chromatic polynomial, independence polynomial, matching polynomial, rank polynomial, and reliability polynomial given by

			(1)
			(2)
			(3)
			(4)
			(5)

where . These have recurrence equations

			(6)
			(7)
			(8)
			(9)
			(10)