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 embedding (Harborth and Möller 1994).

The minimal planar integral embeddings 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 3, graph diameter 3, graph radius 3, and girth 4. The cubical graph is implemented in the Wolfram Language as GraphData["CubicalGraph"].

It is a distance-regular graph with intersection array , and therefore also a Taylor graph.

Its line graph is the cuboctahedral graph.

The maximum number of nodes in a cubical graph that induce a cycle is six (Danzer and Klee 1967; Skiena 1990, p. 149).

A certain construction involving the cubical graph gives an infinite number of connected vertex-transitive graphs that have no Hamilton decomposition (Bryant and Dean 2014).

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

The following table summarizes some properties of the cubical graph.

property	value
automorphism group order	48
characteristic polynomial
chromatic number	2
chromatic polynomial
claw-free	no
clique number	2
graph complement name	8-quartic graph 2
determined by spectrum	yes
diameter	3
distance-regular graph	yes
dual graph name	octahedral graph
edge chromatic number	3
edge connectivity	3
edge count	12
Eulerian	no
girth	4
Hamiltonian	yes
Hamiltonian cycle count	12
Hamiltonian path count	144
integral graph	yes
independence number	4
intersection array
line graph	no
line graph name	cuboctahedral graph
perfect matching graph	no
planar	yes
polyhedral graph	yes
polyhedron embedding names	cube
radius	3
regular	yes
spectrum
square-free	no
traceable	yes
triangle-free	yes
vertex connectivity	3
vertex count	8

REFERENCES

Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 234, 1976.

Bryant, D. and Dean, M. "Vertex-Transitive Graphs that have no Hamilton Decomposition." 25 Aug 2014. 

http://arxiv.org/abs/1408.5211.Danzer, L. and Klee, V. "Lengths of Snakes in Boxes." J. Combin. Th. 2, 258-265, 1967.

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.

Knuth, D. E. The Art of Computer Programming, Volume 4, Fascicle 0: Introduction to Combinatorial Functions and Boolean Functions.. Upper Saddle River, NJ: Addison-Wesley, p. 14, 2008.

Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, p. 266, 1998.

Royle, G. "F008A." http://www.csse.uwa.edu.au/~gordon/foster/F008A.html.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1032, 2002.