The Strongly Regular $(40,12,2,4)$ Graphs

Abstract

In a previous paper it was established that there are at least $27$ non-isomorphic strongly regular $(40,12,2,4)$ graphs. Using a different and more efficient method we have re-investigated these graphs and have now been able to determine them all, and so complete the classification. We have discovered that there are precisely $28$ non-isomorphic $(40,12,2,4)$ strongly regular graphs. The one that was not found in the previous investigation is characterised uniquely by the fact that every neighbour graph is triangle-free.