Klein Graphs


There are a number of graph associated with Felix Klein.

The 24-Klein graph is a weakly regular graph that is the dual graph of the Foster graph F_(056)B. This graph is illustrated above in four order-4 LCF notations.

The 24-Klein graph is distance-regular with intersection array {7,4,1;1,2,7} but is not distance-transitive.

The 24-Klein graph has graph spectrum (-sqrt(7))^8(-1)^7(sqrt(7))^87^1.

The Levi graph of the Klein configuration may be termed the 120-Klein graph.

These graphs are implemented in the Wolfram Language as GraphData["KleinGraph24"] and GraphData["KleinGraph120"], respectively.

See also

Dyck Graph, Foster Graph, Klein configuration, Levi Graph

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Bellarosa, L.; Fowler, P. W.; Lijnen, E.; and Deza, M. "Addition Patterns in Carbon Allotropes: Independence Numbers and d-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, "Klein Graph.", R. B. "Chemical Applications of Topology and Group Theory, 29, Low Density Polymeric Carbon Allotropes Based on Negative Curvature Structures." J. Phys. Chem. 100, 15096-15104, 1996.King, R. B. "Novel Highly Symmetrical Trivalent Graphs Which Lead to Negative Curvature Carbon and Boron Nitride Chemical Structures." Disc. Math. 244, 203-210, 2002.Klein, F. "Über die Transformationen siebenter Ordnung der elliptischen Funktionen." Math. Ann. 14, 428-471, 1879. Reprinted in Gesammelte Mathematische Abhandlungen, 3: Elliptische Funktionen etc. (Ed. R. Fricke et al. ). Berlin: Springer-Verlag, pp. 90-136, 1973.Levy, S. (Ed.). The Eightfold Way: The Beauty of the Klein Quartic. New York: Cambridge University Press, 1999.

Cite this as:

Weisstein, Eric W. "Klein Graphs." From MathWorld--A Wolfram Web Resource.

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