A polyhedral graph corresponding to the skeleton of a Platonic solid. The five platonic graphs, the tetrahedral graph, cubical graph, octahedral graph, dodecahedral graph, and icosahedral graph, are illustrated above. They are special cases of Schlegel graphs.Platonic graphs are graceful (Gardner 1983, pp. 158and 163-164).The following table summarizes the Platonic graphs and some of their properties.graph regularityHamiltonianEulerianvertex-transitiveedge-transitivecubical graphcubic81248yesnoyesyesdodecahedral graphcubic2030120yesnoyesyesicosahedral graphquintic1230120yesnoyesyesoctahedral graphquartic61248yesyesyesyestetrahedral graphcubic4624yesnoyesyes
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..