Balaban 11-cage

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Balaban 11-cage
Balaban 11-cage.svg
The Balaban 11-cage
Named afterAlexandru T. Balaban
Vertices112
Edges168
Radius6
Diameter8
Girth11
Automorphisms64
Chromatic number3
Chromatic index3
PropertiesCubic
Cage
Hamiltonian
Table of graphs and parameters

In the mathematical field of graph theory, the Balaban 11-cage or Balaban (3,11)-cage is a 3-regular graph with 112 vertices and 168 edges named after Alexandru T. Balaban.[1]

The Balaban 11-cage is the unique (3,11)-cage. It was discovered by Balaban in 1973.[2] The uniqueness was proved by Brendan McKay and Wendy Myrvold in 2003.[3]

The Balaban 11-cage is a Hamiltonian graph and can be constructed by excision from the Tutte 12-cage by removing a small subtree and suppressing the resulting vertices of degree two.[4]

It has independence number 52,[5] chromatic number 3, chromatic index 3, radius 6, diameter 8 and girth 11. It is also a 3-vertex-connected graph and a 3-edge-connected graph.

The characteristic polynomial of the Balaban 11-cage is:

[math]\displaystyle{ (x-3) x^{12} (x^2-6)^5 (x^2-2)^{12} (x^3-x^2-4 x+2)^2\cdot }[/math]
[math]\displaystyle{ \cdot(x^3+x^2-6 x-2) (x^4-x^3-6 x^2+4 x+4)^4 \cdot }[/math]
[math]\displaystyle{ \cdot(x^5+x^4-8 x^3-6 x^2+12 x+4)^8 }[/math].

The automorphism group of the Balaban 11-cage is of order 64.[4]

Gallery

References

  1. Weisstein, Eric W.. "Balaban 11-Cage". http://mathworld.wolfram.com/Balaban11-Cage.html. 
  2. Balaban, Alexandru T., Trivalent graphs of girth nine and eleven, and relationships among cages, Revue Roumaine de Mathématiques Pures et Appliquées 18 (1973), 1033-1043. MR0327574
  3. Weisstein, Eric W.. "Cage Graph". http://mathworld.wolfram.com/CageGraph.html. 
  4. 4.0 4.1 Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15 (2008)
  5. (Heal 2016)
  6. P. Eades, J. Marks, P. Mutzel, S. North. "Graph-Drawing Contest Report", TR98-16, December 1998, Mitsubishi Electric Research Laboratories.

References

  • Heal, Maher (2016), "A Quadratic Programming Formulation to Find the Maximum Independent Set of Any Graph", The 2016 International Conference on Computational Science and Computational Intelligence, Las Vegas: IEEE Computer Society