### School of Mathematics and Statistics

### Paul Hafner

University of Auckland

### On the Hoffman-Singleton graph and related graphs

**Friday 6th September, 12-1pm,
Carslaw 373. **

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.