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### Octahedral graph

"The" octahedral graph is the 6-node 12-edge Platonic graph having the connectivity of the octahedron. It is isomorphic to the circulant graph , the cocktail party graph , the complete tripartite graph , and the 4-dipyramidal graph. Several embeddings of this graph are illustrated above.It is implemented in the Wolfram Languageas GraphData["OctahedralGraph"].The octahedral graph has 6 nodes, 12 edges, vertex connectivity 4, edge connectivity 4, graph diameter 2, graph radius 2, and girth 3. It is the unique 6-node quartic graph, and is also a quartic symmetric graph. It has chromatic polynomialand chromatic number 3. It is an integral graph with graph spectrum . Its automorphism group is of order .The octahedral graph is the line graph of the tetrahedralgraph.There are three minimal integral drawings of the octahedral graph, illustrated above, all with maximum edge length of 7 (Harborth and Möller 1994).The..

### Tetrahedral graph

"The" tetrahedral graph is the Platonic graph that is the unique polyhedral graph on four nodes which is also the complete graph and therefore also the wheel graph . It is implemented in the Wolfram Language as GraphData["TetrahedralGraph"].The tetrahedral graph has a single minimal integral drawing, illustrated above (Harborth and Möller 1994), with maximum edge length 4.The minimal planar integral drawing of the tetrahedral graph, illustrated above, has maximum edge length of 17 (Harborth et al. 1987). The tetrahedral graph is also graceful (Gardner 1983, pp. 158 and 163-164).The tetrahedral graph has 4 nodes, 6 edges, vertex connectivity 4, edge connectivity 3, graph diameter 1, graph radius 1, and girth 3. It has chromatic polynomial(1)(2)and chromatic number 4. It is planarand cubic symmetric.The tetrahedral graph is an integral graph with graph spectrum . Its automorphism group has order .The..

### Dodecahedral graph

The dodecahedral graph is the Platonic graph corresponding to the connectivity of the vertices of a dodecahedron, illustrated above in four embeddings. The left embedding shows a stereographic projection of the dodecahedron, the second an orthographic projection, the third is from Read and Wilson (1998, p. 162), and the fourth is derived from LCF notation.It is the cubic symmetric denoted and is isomorphic to the generalized Petersen graph . It can be described in LCF notation as [10, 7, 4, , , 10, , 7, , .It is distance-regular with intersection array and is also distance-transitive.It is also a unit-distance graph (Gerbracht2008), as shown above in a unit-distance drawing.Finding a Hamiltonian cycle on this graph is known as the icosian game. The dodecahedral graph is not Hamilton-connected and is the only known example of a vertex-transitive Hamiltonian graph (other than cycle graphs ) that is not H-*-connected (Stan Wagon, pers...

### Cubical graph

The cubical graph is the Platonic graph corresponding to the connectivity of the cube. It is isomorphic to the generalized Petersen graph , bipartite Kneser graph , 4-crossed prism graph, crown graph , grid graph , hypercube graph , and prism graph . It is illustrated above in a number of embeddings (e.g., Knuth 2008, p. 14).It has 12 distinct (directed) Hamiltonian cycles, corresponding to the unique order-4 LCF notation .It is a unit-distance graph, as shown above in a unit-distance drawing (Harborth and Möller 1994).The minimal planar integral drawings of the cubical graph, illustrated above, has maximum edge length of 2 (Harborth et al. 1987). They are also graceful (Gardner 1983, pp. 158 and 163-164). can be constructed as the graph expansion of with steps 1 and 1, where is a path graph. Excising an edge of the cubical graph gives the prism graph .The cubical graph has 8 nodes, 12 edges, vertex connectivity 3, edge connectivity..

### Icosahedral graph

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..

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