Hall–Janko graph

{{infobox graph

| name = Hall–Janko graph

| image = 300px

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

| namesake = Zvonimir Janko
Marshall Hall

| vertices = 100

| edges = 1800

| automorphisms = 1209600

| diameter = 2

| girth = 3

| radius = 2

| chromatic_number = 10

| chromatic_index =

| 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.{{MathWorld | urlname=Hall-JankoGraph | title=Hall-Janko graph}}

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:Andries E. Brouwer, "[http://www.win.tue.nl/~aeb/drg/graphs/HallJanko.html Hall-Janko graph]".Andries E. Brouwer, "[http://www.win.tue.nl/~aeb/drg/graphs/U3_3.html U₃(3) graph]".

• 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.Robert A. Wilson, 'The Finite Simple Groups', Springer-Verlag (2009), p. 224. 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.