Calculus of functors: Difference between revisions

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rewrite for clarity
 
→Definition: Fixed "such POLYNOMIALS are 'polynomial functors' " to "such FUNCTORS are 'polynomial functors' ". Also changed "THE simplifying condition" to "A simplifying condition".
 
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{{infobox graph
| name = Balaban 10-cage
| image = [[Image:Balaban 10-cage.svg|220px]]
| image_caption = The Balaban 10-cage
| namesake = A. T. Balaban
| vertices = 70
| edges = 105
| automorphisms = 80
| girth = 10
| diameter = 6
| chromatic_number = 2
| chromatic_index = 3
| properties = [[Cubic graph|Cubic]]<br>[[Cage (graph theory)|Cage]]<br>[[Hamiltonian graph|Hamiltonian]]
}}
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Balaban 10-cage''' or '''Balaban (3-10)-cage''' is a 3-[[regular graph]] with 70 vertices and 105 edges named after [[Alexandru Balaban|A. T. Balaban]].<ref>{{MathWorld|urlname=Balaban10-Cage|title=Balaban 10-Cage}}</ref> Published in 1972,<ref>A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1-5. 1972.</ref> It was the first (3-10)-[[cage graph|cage]] discovered but is not unique.<ref>Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. [http://citeseer.ist.psu.edu/448980.html].</ref>  
 
The complete list of (3-10)-cage and the proof of minimality was given by O'Keefe and Wong.<ref>M. O'Keefe and P.K. Wong, A smallest graph of girth 10 and valency 3, J. Combin. Theory Ser. B 29 (1980) 91-105.</ref> There exists 3 distinct (3-10)-cages, the other two being the [[Harries graph]] and the [[Harries-Wong graph]].<ref>Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.</ref> Moreover, the Harries-Wong graph and Harries graph are [[Spectral graph theory|cospectral graphs]].
 
The Balaban 10-cage has chromatic number 2, chromatic index 3, diameter 6, girth 10 and is hamiltonian. It is also a 3-[[k-vertex-connected graph|vertex-connected graph]] and a 3-[[k-edge-connected graph|edge-connected graph]].
 
The [[characteristic polynomial]] of the Balaban 10-cage is : <math>(x-3) (x-2) (x-1)^8 x^2 (x+1)^8 (x+2) (x+3) (x^2-6)^2 (x^2-5)^4 (x^2-2)^2 (x^4-6 x^2+3)^8</math>.
 
==Gallery==
<gallery>
Image:balaban_10-cage_2COL.svg|The [[chromatic number]] of the Balaban 10-cage is&nbsp;2.
Image:balaban_10-cage_3color_edge.svg|The [[chromatic index]] of the Balaban 10-cage is&nbsp;3.
Image: balaban_10-cage_alternative_drawing.svg|Alternative drawing of the Balaban 10-cage.
</gallery>
==See also==
[[Molecular graph]]
 
== References ==
{{reflist}}
 
{{DEFAULTSORT:Balaban 10-Cage}}
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Latest revision as of 08:47, 5 November 2013

Template:Infobox graph In the mathematical field of graph theory, the Balaban 10-cage or Balaban (3-10)-cage is a 3-regular graph with 70 vertices and 105 edges named after A. T. Balaban.[1] Published in 1972,[2] It was the first (3-10)-cage discovered but is not unique.[3]

The complete list of (3-10)-cage and the proof of minimality was given by O'Keefe and Wong.[4] There exists 3 distinct (3-10)-cages, the other two being the Harries graph and the Harries-Wong graph.[5] Moreover, the Harries-Wong graph and Harries graph are cospectral graphs.

The Balaban 10-cage has chromatic number 2, chromatic index 3, diameter 6, girth 10 and is hamiltonian. It is also a 3-vertex-connected graph and a 3-edge-connected graph.

The characteristic polynomial of the Balaban 10-cage is : (x−3)(x−2)(x−1)8x2(x+1)8(x+2)(x+3)(x2−6)2(x2−5)4(x2−2)2(x4−6x2+3)8.

Gallery

See also

Molecular graph

References

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  2. ↑ A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1-5. 1972.
  3. ↑ Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. [1].
  4. ↑ M. O'Keefe and P.K. Wong, A smallest graph of girth 10 and valency 3, J. Combin. Theory Ser. B 29 (1980) 91-105.
  5. ↑ Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.
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