Hall-Janko Graph

The Hall-Janko graph, also known as the Hall-Janko-Wales graph, is a strongly regular graph on 100 nodes with parameters . It is also a distance-regular graph with intersection array , as well as being distance-transitive. It is illustrated above in an embedding due to C. Rocchini.

It is an integral graph with graph spectrum . The Hall-Janko graph has independence number 10 and chromatic number 10, while the graph complement of the Hall-Janko graph has independence number 4 and chromatic number 25 (Brouwer).

The Janko group J₂ acts on the graph, realizing it as a subgroup of the full automorphism group (Hall and Wales 1968). The full group is denoted and has group order (DistanceRegular.org). Since has group order , it has index 2 in the full group.

It is implemented in the Wolfram Language as GraphData["HallJankoGraph"].

It is the middle graph in the chain of local graphs known as the Suzuki tower.

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More things to try:

Bagchi, B. "A Regular Two-Graph Admitting the Hall-Janko-Wales Group." Combinatorial Mathematics and Applications (Calcutta, 1988). Sankhyā Ser. A 54, 35-45, 1992.Brouwer, A. E. "A Construction of the HJ Graph." Preprint. 1989.Brouwer, A. E. "Hall-Janko Graph." https://aeb.win.tue.nl/drg/graphs/HallJanko.html.DistanceRegular.org. "Hall-Janko Graph." https://www.math.mun.ca/distanceregular/graphs/halljanko.html.Hall, M. Jr. and Wales, D. "The Simple Group of Order 604,800." J. Algebra 9, 417-450, 1968.House of Graphs. "Hall Janko Graph." https://houseofgraphs.org/graphs/1140.Jørgensen, L. K. and Klin, M. "Switching of Edges in Strongly Regular Graphs. I. A Family of Partial Difference Sets on 100 Vertices." Electr. J. Combin. 10, No. R17, 2003.