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.

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:

The characteristic polynomial of the Hall–Janko graph is ${\displaystyle (x-36)(x-6)^{36}(x+4)^{63}}$. Therefore the Hall–Janko graph is an integral graph: its spectrum consists entirely of integers.