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### 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..

### Complete Graph

A complete graph is a graph in which each pair of graph vertices is connected by an edge. The complete graph with graph vertices is denoted and has (the triangular numbers) undirected edges, where is a binomial coefficient. In older literature, complete graphs are sometimes called universal graphs.The complete graph is also the complete n-partite graph .The complete graph on nodes is implemented in the Wolfram Language as CompleteGraph[n]. Precomputed properties are available using GraphData["Complete", n]. A graph may be tested to see if it is complete in the Wolfram Language using the function CompleteGraphQ[g].The complete graph on 0 nodes is a trivial graph known as the null graph, while the complete graph on 1 node is a trivial graph known as the singleton graph.In the 1890s, Walecki showed that complete graphs admit a Hamilton decomposition for odd , and decompositions into Hamiltonian cycles plus a perfect matching for..

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