The icosahedral graph is the Platonic graph whose nodes have the connectivity of the icosahedron, illustrated above in a number of embeddings. The icosahedral graph has 12 vertices and 30 edges.Since the icosahedral graph is regular and Hamiltonian, it has a generalized LCF notation. In fact, there are two distinct generalized LCF notations of order 6-- and --8 of order 2, and 17 of order 1, illustrated above.It is implemented in the Wolfram Languageas GraphData["IcosahedralGraph"].It is a distance-regular graph with intersection array , and therefore also a Taylor graph. It is also distance-transitive.There are two minimal integral drawings of the icosahedral graph, illustrated above, all with maximum edge length of 8 (Harborth and Möller 1994). It is also graceful (Gardner 1983, pp. 158 and 163-164; Gallian 2018, p. 35), with five fundamentally different labelings (Gardner 1983, p. 164).The..