Algebra Seminar

The Hoffman-Singleton graph is the unique Moore graph of order 50, degree 7, diameter 2 and girth 5. In 1998, three infinite families of graphs, modelled on the Hoffman-Singleton graph, were constructed by McKay, Miller and Siran, using voltage assignments.

In the course of studying these graphs it emerged that Robertson's pentagon-pentagram construction of the Hoffman-Singleton graph should be seen as describing the incidence graph of a bi-affine plane with some additional edges. The `same' description applies to the graphs of McKay-Miller-Siran and is instrumental in the determination of their automorphism groups.

We will describe these developments, concentrating on the example of the Hoffman-Singleton graph. We will also look at the geometric interpretation of 15-cocliques in the Hoffman-Singleton graph which are important in the construction of the Higman-Sims graph.