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 graph vertices of the outer cycle (Gallian 2018). The gear graph  has  nodes 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 embeddings shown above.

Attractive derived unit-distance graph are produced by taking the vertex sets from the matchstick embeddings and connecting all pairs of vertices separate 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 cycle of the wheel, the resulting graph is also graceful (Gallian 2018).

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)

REFERENCES

Brandstädt, A.; Le, V. B.; and Spinrad, J. P. Graph Classes: A Survey. Philadelphia, PA: SIAM, p. 19, 1987.

Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018.

 https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Ma, K. J. and Feng, C. J. "On the Gracefulness of Gear Graphs." Math. Practice Theory, No. 4, 72-73, 1984.

Liu, Y. "The Gracefulness of the Star Graph with Top Sides." J. Sichuan Normal Univ. 18, 52-60, 1995.

Liu, Y. "Crowns Graphs  Are Harmonious Graphs." Hunan Annals Math. 16, 125-128, 1996.