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Generalized Hexagon


A generalized hexagon is a generalized polygon of order 6.

GH(1,2) is more commonly known as the Heawood graph, but is also the (3,6)-cage graph, the cubic vertex-transitive graph Ct15, the cubic symmetric graph F_(014)A, the 69-Haar graph, and is an incidence graph of a 2-(7,3,1) design.

GH(1,3) is the (4,6)-cage graph, 4137-Haar graph, and is an incidence graph of a 2-(13,4,1) design.

GH(2,1) is the Bouwer graph B(2,3,7), line graph of the Heawood graph, and is a distance-regular graph with intersection array {4,2,2;1,1,2},

GH(3,1) is the line graph of the (4,6)-cage, is also known as the flag graph of PG(2,3) (DistanceRegular.org), and is a distance-regular graph with intersection array {6,3,3;1,1,2}.

GH(4,1) is the line graph of the (5,6)-cage, is also known as the flag graph of PG(2,4) (DistanceRegular.org), and is the distance-regular graph with intersection array {8,4,4;1,1,2}.

The generalized hexagons GH(t,s) are line graphs of the generalized hexagons GH(s,t).

The following table summarizes some generalized hexagons.

graphVother namesincidencegraph spectrum
GH(1, 2)14Heawood graph(7,3,1)(-3)^1(-sqrt(2))^6(sqrt(2))^63^1
GH(1, 3)26(4, 6)-cage graph, incidence graph of PG(2,3)(13,4,1)(-4)^1(-sqrt(3))^(12)(sqrt(3))^(12)4^1
GH(1, 4)42(5, 6)-cage graph(21,5,1)(-5)^1(-2)^(20)2^(20)5^1
GH(1, 5)62(6, 6)-cage graph(31,6,1)(-6)^1(-sqrt(5))^(30)(sqrt(5))^(30)6^1
GH(1, 7)114(8, 6)-cage graph(57,8,1)(-8)^1(-sqrt(7))^(56)(sqrt(7))^(56)8^1
GH(1, 8)146(9, 6)-cage graph(73,9,1)(-9)^1(-2sqrt(2))^(72)(2sqrt(2))^(72)9^1
GH(1, 9)182(10, 6)-cage graph(91,10,1)(-10)^1(-3)^(90)3^(90)10^1
GH(2, 1)21(2,3,7)-Bouwer graph, flag graph of PG(2,2)(-2)^8(1-sqrt(2))^6(1+sqrt(2))^64^1
GH(2, 8)819(-9)^(26)(-3)^(468)5^(324)18^1
GH(3, 1)52(-2)^(27)(2-sqrt(3))^(12)(2+sqrt(3))^(12)6^1
GH(4, 1)105(-2)^(64)1^(20)5^(20)8^1
GH(5, 1)186(-2)^(125)(4-sqrt(5))^(30)(4+sqrt(5))^(30)10^1
GH(7, 1)456(-2)^(343)(6-sqrt(7))^(56)(6+sqrt(7))^(56)14^1
GH(8, 1)657(-2)^(512)(7-sqrt(8))^(72)(7+sqrt(8))^(72)16^1
GH(8, 2)2457(-3)^(1664)3^(468)11^(324)24^1

See also

Cage Graph, Generalized Dodecagon, Generalized Octagon, Generalized Polygon, Generalized Quadrangle

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References

Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, p. 204, 1989.Brouwer, A. and Koolen, J. "The Distance-Regular Graphs of Valency Four." J. Algebraic Combin. 10, 5-24, 1999.DistanceRegular.org. "Flag Graph of PG(2,3)." http://www.distanceregular.org/graphs/flag-pg2.3.html.DistanceRegular.org. "Flag Graph of PG(2,4)." http://www.distanceregular.org/graphs/flag-pg2.4.html.DistanceRegular.org. "Point Graphs of GH(2,2) and its Dual." http://www.distanceregular.org/graphs/point-gh2.2.html.Godsil, C. and Royle, G. "Two Generalized Hexagons." §5.7 in Algebraic Graph Theory. New York: Springer-Verlag, pp. 88-90, 2001.van Dam, E. R. and Haemers, W. H. "Which Graphs Are Determined by Their Spectrum?" Lin. Algebra Appl. 373, 139-162, 2003.

Referenced on Wolfram|Alpha

Generalized Hexagon

Cite this as:

Weisstein, Eric W. "Generalized Hexagon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeneralizedHexagon.html

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