In the mathematical field of graph theory, an Archimedean graph is a graph that forms the skeleton of one of the Archimedean solids. There are 13 Archimedean graphs, and all of them are regular, polyhedral (and therefore by necessity also 3-vertex-connected planar graphs), and also Hamiltonian graphs.[1]

Along with the 13, the set of infinite prism graphs and antiprism graphs can also be considered Archimedean graphs.[2]

Graph elements
Name Graph Degree Edges Vertices Order
truncated tetrahedral graph 3 18 12 24
cuboctahedral graph 4 24 12 48
truncated cubical graph 3 36 24 48
truncated octahedral graph 3 36 24 48
rhombicuboctahedral graph 4 48 24 48
truncated cuboctahedral graph
(great rhombicuboctahedron)
3 72 48 48
snub cubical graph 5 60 24 24
icosidodecahedral graph 4 60 30 120
truncated dodecahedral graph 3 90 60 120
truncated icosahedral graph 3 90 60 120
rhombicosidodecahedral graph 4 120 60 120
truncated icosidodecahedral graph
(great rhombicosidodecahedron)
3 180 120 120
snub dodecahedral graph 5 150 60 60

Platonic graph
Wheel graph

References

An Atlas of Graphs, p. 267-270

An Atlas of Graphs, p. 261

Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp. 261, 267-269.