mathworld.wolfram.com

Dodecahedral Graph -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorld

DodecahedralGraphEmbeddings

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.

The dodecahedral graph is the skeleton of the great stellated dodecahedron as well as the dodecahedron.

It is the cubic symmetric denoted F_(020)A and is isomorphic to the generalized Petersen graph GP(10,2). It can be described in LCF notation as [10, 7, 4, -4, -7, 10, -4, 7, -7, 4]^2.

The dodecahedral graph is implemented in the Wolfram Language as GraphData["DodecahedralGraph"].

It is distance-regular with intersection array {3,2,1,1,1;1,1,1,2,3} and is also distance-transitive.

DodecahedralGraphUnitDistance

It is also a unit-distance graph (Gerbracht 2008), as shown above in a unit-distance embedding.

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 C_n) that is not H-*-connected (Stan Wagon, pers. comm., May 20, 2013).

The dodecahedral graph has 20 nodes, 30 edges, vertex connectivity 3, edge connectivity 3, graph diameter 5, graph radius 5, and girth 5. Its has chromatic number 3. Its graph spectrum is Spec(G)=(-sqrt(5))^3(-2)^40^41^5(sqrt(5))^33^1 (Buekenhout and Parker 1998; Cvetkovic et al. 1998, p. 308). Its automorphism group is of order |Aut(G)|=120 (Buekenhout and Parker 1998).

DodecahedralGraphMinimalPlanarIntegralDrawing

The minimal planar integral embedding of the dodecahedral graph has maximum edge length of 2 (Harborth et al. 1987). It is also graceful (Gardner 1983, pp. 158 and 163-164; Gallian 2018, p. 35; Knuth 2024) with 784298856 fundamentally different labelings, giving a total number of 2×120×784298856=188231725440 graceful labelings (B. Dobbelaere, pers. comm., Oct. 22, 2020), a number independently (and nearly simultaneously!) determined by T. Rokicki on Oct. 6, 2020 (D. Knuth, pers. comm., Jul. 6, 2023).

The dodecahedral graph can be constructed as the graph expansion of 10P_2 with steps 1 and 2, where P_2 is a path graph (Biggs 1993, p. 119).

The skeleton of the great stellated dodecahedron is isomorphic to the dodecahedral graph.

The line graph of the dodecahedral graph is the icosidodecahedral graph. The graph square of the dodecahedral graph is the crossed dodecahedral graph.

The dodecahedral graph has chromatic polynomial

pi(z)=(z-2)(z-1)z(z^(17)-27z^(16)+352z^(15)-2950z^(14)+17839z^(13)-82777z^(12)+305866z^(11)-921448z^(10)+2297495z^9-4783425z^8+8347700z^7-12195590z^6+14808795z^5-14713381z^4+11613602z^3-6892084z^2+2751604z-555984).

DodecahedralGraphMatrices

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

The bipartite double graph of the dodecahedral graph is the cubic symmetric graph F_(040)A.

The following table summarizes properties of the dodecahedral graph.

See also

Cubic Symmetric Graph, Cubical Graph, Grünbaum Graphs, Icosahedral Graph, Icosian Game, Octahedral Graph, Platonic Graph, Tetrahedral Graph

Explore with Wolfram|Alpha

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987.Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 234, 1976.Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension <=4." Disc. Math. 186, 69-94, 1998.Chartrand, G. Introductory Graph Theory. New York: Dover, 1985.Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.DistanceRegular.org. "Dodecahedron." http://www.distanceregular.org/graphs/dodecahedron.html.Gardner, M. "Golomb's Graceful Graphs." Ch. 15 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 152-165, 1983.Gerbracht, E. H.-A. "On the Unit Distance Embeddability of Connected Cubic Symmetric Graphs." Kolloquium über Kombinatorik. Magdeburg, Germany. Nov. 15, 2008.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.Knuth, D. E. Problem 101 in §7.2.2.3 in The Art of Computer Programming, Vol. 4. Pre-Fascicle 7A, Dec. 5, pp. 122 and 181-192, 2024.Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, p. 266, 1998.Royle, G. "F020A." http://www.csse.uwa.edu.au/~gordon/foster/F020A.html.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 198, 1990.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1032, 2002.

Referenced on Wolfram|Alpha

Dodecahedral Graph

Cite this as:

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

Subject classifications