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Klein Graph


KleinGraph

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

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

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

The Klein graph is implemented in the Wolfram Language as GraphData["KleinGraph"].

Its properties are summarized in the following table.

propertyvalue
automorphism group order336
characteristic polynomial(x-7)(x+1)^7(x^2-7)^8
chromatic number4
chromatic polynomial?
claw-freeno
clique number3
graph complement name?
cospectral graph names-
determined by spectrumno
diameter3
distance-regular graphyes
dual graph namecubic symmetric graph F_(056)B
edge chromatic number7
edge connectivity7
edge count84
edge transitiveyes
Eulerianno
girth3
Hamiltonianyes
Hamiltonian cycle count?
Hamiltonian path count?
integral graphno
independence number6
line graphno
line graph name?
perfect matching graphno
planarno
polyhedral graphno
radius3
regularyes
square-freeno
symmetricyes
traceableyes
triangle-freeno
vertex connectivity7
vertex count24
vertex transitiveyes
weakly regular parameters(24,(7),(2),(0,2))

See also

Dyck Graph

Explore with Wolfram|Alpha

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." http://www.distanceregular.org/graphs/klein.html.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: 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 Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KleinGraph.html

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