"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..