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Rhombic Dodecahedral Graph


RhombicDodecahedralGraph

The rhombic dodecahedral graph is the Archimedean dual graph which is the skeleton of the rhombic dodecahedron (as well as the Bilinski dodecahedron). It is the Levi graph of the Miquel configuration. The rhombic dodecahedral graph is bipartite, edge-transitive, nonhamiltonian, planar, polyhedral, and untraceable. It is illustrated above in a number of embeddings.

The graph was rediscovered by A. Fruchard for its property of being small (14 vertices), polyhedral, and untraceable. For this reason, it was termed the "Fruchard graph" by Maddaloni and Zamfirescu (2016) and van Cleemput and Zamfirescu (2018), apparently without realizing its origin as the skeleton of the rhombic dodecahedron. The rhombic dodecahedral graph is however not alone in having these properties; the small triakis octahedral graph is another 14-vertex polyhedral untraceable graph.

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

RhombicDodecahedralGraphMatrices

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

The following table summarizes some properties of the graph.

propertyvalue
automorphism group order48
characteristic polynomial(x-2)^3x^6(x+2)^3(x^2-12)
chromatic number2
chromatic polynomial(x-1)x(x^(12)-23x^(11)+253x^(10)-1759x^9+8615x^8-31361x^7+87205x^6-187127x^5+308232x^4-380364x^3+333138x^2-184963x+48831)
claw-freeno
clique number2
determined by spectrum?
diameter4
distance-regular graphno
dual graph namecuboctahedral graph
edge chromatic number4
edge connectivity3
edge count24
Eulerianno
girth4
Hamiltonianno
Hamiltonian cycle count0
Hamiltonian path count0
integral graphno
independence number8
line graph?
perfect matching graphno
planaryes
polyhedral graphyes
polyhedron embedding namesrhombic dodecahedron
radius4
regularno
square-freeno
traceableno
triangle-freeyes
vertex connectivity3
vertex count14

See also

Archimedean Dual Graph, Rhombic Dodecahedron

Explore with Wolfram|Alpha

References

Maddaloni, A. and Zamfirescu, C. T. "A Cut Locus for Finite Graphs and the Farthest Point Mapping." Disc. Math. 339, 354-364, 2016.van Cleemput, N. and Zamfirescu, C. T. "Regular Non-Hamiltonian Polyhedral Graphs." Appl. Math. Comput. 338 192-206, 2018.

Referenced on Wolfram|Alpha

Rhombic Dodecahedral Graph

Cite this as:

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

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