Today I watched, on Natalia Maslova’s on-line seminar from Yekaterinburg, a talk by Sergey Shpectorov from Birmingham, on the non-existence of a strongly regular graph with parameters (84,14,3,2): this is a graph with 84 vertices, regular with valency 14, and with two vertices having three or two neighbours according as whether they are adjacent or not.

What was remarkable about the proof is the hugeness of the computation involved: Sergey used 96 cores at the University of Birmingham for well over a year to get the result. This is not how I prefer to prove theorems, but I take off my hat to Sergey for having done it!

The strategy was to analyse the trillions of configurations for a 30-vertex subgraph consisting of the neighbourhood of a maximal clique of size 3, to decide whether the image under projection onto one of the eigenspaces of the adjacency matrix is positive semidefinite (as it must be if the graph exists). This eliminated all possibilities except a very small number (in double figures) which required further analysis.

A lot of clever trickery went into the proof; some of it, I admit, I didn’t really understand. But an impressive piece of work anyway.

In a way I was reminded of something that happened during the classification of finite simple groups. Charles Sims was reported to have said, at some point, that if he were given a million dollars, he could construct the Lyons group. In the end he constructed it much more cheaply than that!

About Peter Cameron

I count all the things that need to be counted.