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]

Balaban 11-cage
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

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:

.

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

Gallery edit

References edit

  1. ^ Weisstein, Eric W. "Balaban 11-Cage". MathWorld.
  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". MathWorld.
  4. ^ a b 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 edit

  • 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