# Hall–Janko graph

Hall–Janko graph | |
---|---|

HJ as Foster graph (90 outer vertices) plus Steiner system S(3,4,10) (10 inner vertices). | |

Named after | Zvonimir Janko Marshall Hall |

Vertices | 100 |

Edges | 1800 |

Radius | 2 |

Diameter | 2 |

Girth | 3 |

Automorphisms | 1209600 |

Chromatic number | 10 |

Properties | Strongly regular Vertex-transitive Cayley graph Eulerian Hamiltonian Integral |

Table of graphs and parameters |

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]}

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 U_{3}(3), the simple group of order 6048:^{[2]}^{[3]}

- In U
_{3}(3) there are 36 simple maximal subgroups of order 168. These are the vertices of a subgraph, the U_{3}(3) graph. A 168-subgroup has 14 maximal subgroups of order 24, isomorphic to S_{4}. Two 168-subgroups are called adjacent when they intersect in a 24-subgroup. The U_{3}(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 U
_{3}(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[edit]

**^**Weisstein, Eric W. "Hall-Janko graph".*MathWorld*.**^**Andries E. Brouwer, "Hall-Janko graph".**^**Andries E. Brouwer, "U_{3}(3) graph".**^**Robert A. Wilson, 'The Finite Simple Groups', Springer-Verlag (2009), p. 224.