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
Cayley 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]

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]

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


  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.
This article is issued from Wikipedia - version of the 4/11/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.