|Named after||Alan J. Hoffman
Robert R. Singleton
In the mathematical field of graph theory, the Hoffman–Singleton graph is a 7-regular undirected graph with 50 vertices and 175 edges. It is the unique strongly regular graph with parameters (50,7,0,1). It was constructed by Alan Hoffman and Robert Singleton while trying to classify all Moore graphs, and is the highest order Moore graph known to exist. Since it is a Moore graph where each vertex has degree 7, and the girth is 5, it is a (7,5)-cage.
There are many ways to construct the Hoffman-Singleton graph.
Construction from pentagons and pentagrams
Take five pentagons Ph and five pentagrams Qi, so that vertex j of Ph is adjacent to vertices j-1 and j+1 of Ph and vertex j of Qi is adjacent to vertices j-2 and j+2 of Qi. Now join vertex j of Ph to vertex hi+j of Qi. (All indices mod 5.)
Construction from triads and Fano planes
Take the Fano plane and permute its 7 points to make a set of 30 Fanos. Pick any one of these 30 Fanos; there will be another 14 Fanos that share exactly one triplet ("line") with the first one. Take those 15 Fanos and discard the other 15. Take the 7C3 = 35 triads on 7 numbers. Now connect a triad to a Fano that includes it, and connect disjoint triads to each other. The resulting graph is the Hoffman-Singleton graph, with the 50 vertices corresponding to the 35 triads + 15 Fanos, and each vertex has degree 7. Vertices corresponding to Fanos are linked to 7 triads by definition, as the Fano plane has 7 lines. Each triad is linked to 3 different Fanos that include it, and to 4 other triads with which it is disjoint.
The automorphism group of the Hoffman–Singleton graph is a group of order 252,000 isomorphic to PΣU(3,52) the semidirect product of the projective special unitary group PSU(3,52) with the cyclic group of order 2 generated by the Frobenius automorphism. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Hoffman–Singleton graph is a symmetric graph. The stabilizer of a vertex of the graph is isomorphic to the symmetric group S7 on 7 letters. The setwise stabilizer of an edge is isomorphic to Aut(A6)=A6.22, where A6 is the alternating group on 6 letters. Both of the two types of stabilizers are maximal subgroups of the whole automorphism group of the Hoffman-Singleton graph.
Using only the fact that the Hoffman–Singleton graph is a strongly regular graph with parameters (50,7,0,1), it can be shown that there are 1260 5-cycles contained in the Hoffman–Singleton graph.
- Weisstein, Eric W., "Hoffman-Singleton Graph", MathWorld.
- Hafner, P. R. "The Hoffman-Singleton Graph and Its Automorphisms." J. Algebraic Combin. 18, 7-12, 2003.
- Royle, G. "Re: What is the Edge Chromatic Number of Hoffman-Singleton?" GRAPHNET@istserv.nodak.edu posting. Sept. 28, 2004. 
- Conder, M.D.E.; Stokes, K.: "Minimum genus embeddings of the Hoffman-Singleton graph", preprint, August 2014.
- Brouwer, Andries E., Hoffman-Singleton graph.
- Hoffman, Alan J.; Singleton, Robert R. (1960), "Moore graphs with diameter 2 and 3" (PDF), IBM Journal of Research and Development 5 (4): 497–504, doi:10.1147/rd.45.0497, MR 0140437.
- Wong, Pak-Ken. "On the uniqueness of the smallest graph of girth 5 and valency 6". Journal of Graph Theory 3: 407—409.
- Benson, C. T.; Losey, N. E. (1971), "On a graph of Hoffman and Singleton", Journal of Combinatorial Theory. Series B 11 (1): 67–79, doi:10.1016/0095-8956(71)90015-3, ISSN 0095-8956, MR 0281658
- Holton, D.A.; Sheehan, J. (1993), The Petersen graph, Cambridge University Press, pp. 186 and 190, ISBN 0-521-43594-3.