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

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 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 $(x-36)(x-6)^{36}(x+4)^{63}$. 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.