The Klein graph is a weakly regular graph that is the dual graph of the cubic
symmetric graph 
.
The Klein graph is illustrated above in four order-4 LCF
notations.
The Klein graph is distance-regular with intersection array 
but is not distance-transitive.
The Klein graph has graph spectrum 
.
The Klein graph is implemented in the Wolfram Language as GraphData["KleinGraph"].
Its properties are summarized in the following table.
| property | value |
| automorphism group order | 336 |
| characteristic polynomial |  |
| chromatic number | 4 |
| chromatic polynomial | ? |
| claw-free | no |
| clique number | 3 |
| graph complement name | ? |
| cospectral graph names | - |
| determined by spectrum | no |
| diameter | 3 |
| distance-regular graph | yes |
| dual graph name | cubic symmetric graph  |
| edge chromatic number | 7 |
| edge connectivity | 7 |
| edge count | 84 |
| edge transitive | yes |
| Eulerian | no |
| girth | 3 |
| Hamiltonian | yes |
| Hamiltonian cycle count | ? |
| Hamiltonian path count | ? |
| integral graph | no |
| independence number | 6 |
| line graph | no |
| line graph name | ? |
| perfect matching graph | no |
| planar | no |
| polyhedral graph | no |
| radius | 3 |
| regular | yes |
| square-free | no |
| symmetric | yes |
| traceable | yes |
| triangle-free | no |
| vertex connectivity | 7 |
| vertex count | 24 |
| vertex transitive | yes |
| weakly regular parameters |  |
-Codes in the Klein and Related Graphs." J. Chem. Inf.
Comput. Sci. 44, 1314-1323, 2004.Ceulemans, A.; King, R. B.;
Bovin, S. A.; Rogers, K. M.; Troisi, A.; and Fowler, P. W. "The
Heptakisoctahedral Group and Its Relevance to Carbon Allotropes with Negative Curvature."
J. Math. Chem. 26, 101-123, 1999.DistanceRegular.org.
"Klein Graph."