Hall–Janko graph

In the mathematical field of graph theory, the Hall–Janko graph, also known as the Hall-Janko-Wales graph, is a 36-regular undirected graph with 100 vertices and 1800 edges.[1]

Hall–Janko graph
HJ as Foster graph (90 outer vertices) plus Steiner system S(3,4,10) (10 inner vertices).
Named afterZvonimir Janko
Marshall Hall
Vertices100
Edges1800
Radius2
Diameter2
Girth3
Automorphisms1209600
Chromatic number10
PropertiesStrongly regular
Vertex-transitive
Cayley graph
Eulerian
Hamiltonian
Integral
Table of graphs and parameters

It is a rank 3 strongly regular graph with parameters (100,36,14,12) and a maximum coclique of size 10. This parameter set is not unique, it is however uniquely determined by its parameters as a rank 3 graph. The Hall–Janko graph was originally constructed by D. Wales to establish the existence of the Hall-Janko group as an index 2 subgroup of its automorphism group.

The Hall–Janko graph can be constructed out of objects in U3(3), the simple group of order 6048:[2][3]

  • In U3(3) there are 36 simple maximal subgroups of order 168. These are the vertices of a subgraph, the U3(3) graph. A 168-subgroup has 14 maximal subgroups of order 24, isomorphic to S4. Two 168-subgroups are called adjacent when they intersect in a 24-subgroup. The U3(3) graph is strongly regular, with parameters (36,14,4,6)
  • There are 63 involutions (elements of order 2). A 168-subgroup contains 21 involutions, which are defined to be neighbors.
  • Outside U3(3) let there be a 100th vertex C, whose neighbors are the 36 168-subgroups. A 168-subgroup then has 14 common neighbors with C and in all 1+14+21 neighbors.
  • An involution is found in 12 of the 168-subgroups. C and an involution are non-adjacent, with 12 common neighbors.
  • Two involutions are defined as adjacent when they generate a dihedral subgroup of order 8.[4] An involution has 24 involutions as neighbors.

The characteristic polynomial of the Hall–Janko graph is . Therefore the Hall–Janko graph is an integral graph: its spectrum consists entirely of integers.

References
  1. Weisstein, Eric W. "Hall-Janko graph". MathWorld.
  2. Andries E. Brouwer, "Hall-Janko graph".
  3. Andries E. Brouwer, "U3(3) graph".
  4. Robert A. Wilson, 'The Finite Simple Groups', Springer-Verlag (2009), p. 224.
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