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Tetrahedral Graph


TetrahedralGraphEmbeddings

"The" tetrahedral graph is the Platonic graph that is the unique polyhedral graph on four nodes which is also the complete graph K_4 and therefore also the wheel graph W_4. It is implemented in the Wolfram Language as GraphData["TetrahedralGraph"].

TetrahedralGraphMinimalEmbedding

The tetrahedral graph has a single minimal integral embedding, illustrated above (Harborth and Möller 1994), with maximum edge length 4.

TetrahedralGraphMinimalPlanarIntegralDrawing

The minimal planar integral embedding 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

pi_G(z)=z(z-1)(z-2)(z-3)
(1)
=z^4-6z^3+11z^2-6z
(2)

and chromatic number 4. It is planar and cubic symmetric.

The tetrahedral graph is an integral graph with graph spectrum Spec(G)=(-1)^33^1. Its automorphism group has order |Aut(G)|=24.

The tetrahedral graph is the line graph of the star graph S_5, and the line graph of the tetrahedral graph is the octahedral graph.

TetrahedralGraphMatrices

The plots above show the adjacency, incidence, and graph distance matrices for the tetrahedral graph.

The bipartite double graph of the tetrahedral graph is the cubical graph.

The following table summarizes some properties of the tetrahedral graph.

propertyvalue
automorphism group order24
characteristic polynomial(x-3)(x+1)^3
chromatic number4
chromatic polynomial(x-3)(x-2)(x-1)x
circulant graphCi_4(1,2)
claw-freeyes
clique number4
graph complement name4-empty graph
determined by spectrumyes
diameter1
distance-regular graphyes
dual graph nametetrahedral graph
edge chromatic number3
edge connectivity3
edge count6
Eulerianno
girth3
Hamiltonianyes
Hamiltonian cycle count6
Hamiltonian path count24
integral graphyes
independence number1
intersection array{3;1}
LCF notation[-2]^4
line graphyes
line graph nameoctahedral graph
perfect matching graphno
planaryes
polyhedral graphyes
polyhedron embedding namestetrahedron
radius1
regularyes
spectrum(-1)^33^1
square-freeno
strongly regular parameters(4,3,2,0)
traceableyes
triangle-freeno
vertex connectivity3
vertex count4

More generally, a Johnson graph of the from J(n,3) (with n>=6) is known as an n-tetrahedral graph. For clarity, this work calls such graphs tetrahedral Johnson graphs.


See also

Cubic Symmetric Graph, Cubical Graph, Dodecahedral Graph, Icosahedral Graph, Integral Graph, Octahedral Graph, Platonic Graph, Polyhedral Graph, Tetrahedral Johnson Graph, Tetrahedron

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References

Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 234, 1976.Gardner, M. "Golomb's Graceful Graphs." Ch. 15 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 152-165, 1983.Harborth, H. and Möller, M. "Minimum Integral Drawings of the Platonic Graphs." Math. Mag. 67, 355-358, 1994.Harborth, H.; Kemnitz, A.; Möller, M.; and Süssenbach, A. "Ganzzahlige planare Darstellungen der platonischen Körper." Elem. Math. 42, 118-122, 1987.Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, p. 266, 1998.Royle, G. "F004A." http://www.csse.uwa.edu.au/~gordon/foster/F004A.html.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1032, 2002.

Cite this as:

Weisstein, Eric W. "Tetrahedral Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TetrahedralGraph.html

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