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Klein Graphs

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The name "Klein graph" is used for a number of graphs associated with Felix Klein.

KleinGraph24

The 24-Klein graph is a 7-regular graph with 24 vertices and 84 edges. It is the dual graph of the 56-Klein graph in the Klein map. This weakly regular graph is illustrated above in a number of drawings.

The 24-Klein graph is distance-regular with intersection array {7,4,1;1,2,7} but is not distance-transitive. It has graph spectrum (-sqrt(7))^8(-1)^7(sqrt(7))^87^1.

The 56-Klein graph, also known as the Foster census graph F_(056)B, is the skeleton of the Klein map {7,3}_8. It is a cubic graph with 56 vertices, 84 edges, and girth 7. The 24 heptagonal faces define a cellular embedding of the graph on an orientable surface of genus 3 (McCooey 2009).

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

The 24-, 56-, and 120-Klein graphs are implemented in the Wolfram Language as GraphData["KleinGraph24"], GraphData["Foster056B"], and GraphData["KleinGraph120"], respectively.

See also

Dyck Graph, Klein Configuration, Klein Map, Klein Quartic, Levi Graph, Regular Map

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References

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, 1999.DistanceRegular.org. "Klein Graph." https://www.math.mun.ca/distanceregular/graphs/klein.html.House of Graphs. Klein Graphs. Klein Distance 2 Graph, Klein Graph, Graph 36660, and Klein Graph 120.King, 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, Germany: Springer-Verlag, pp. 90-136, 1973.Levy, S. (Ed.). The Eightfold Way: The Beauty of the Klein Quartic. New York: Cambridge University Press, 1999.McCooey, D. I. "A Non-Self-Intersecting Polyhedral Realization of the All-Heptagon Klein Map." Symmetry: Culture and Science 20, 247-268, 2009. https://journal-scs.symmetry.hu/content-pages/volume-20-numbers-1-4-pages-001-448-2009/.

Referenced on Wolfram|Alpha

Klein Graphs

Cite this as:

Weisstein, Eric W. "Klein Graphs." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KleinGraphs.html

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