Strongly regular graphs with no triangles

Mikhail Klin (Ben-Gurion University of the Negev, Israel)

A strongly regular graph (briefly SRG) $\Gamma$ is called primitive if both $\Gamma$ and its complementary graph are connected. Only primitive SRGs are considered. The most famous SRG with no triangles has parameters (1100, 22, 0, 6), it was described in the paper by Higman and Sims in 1968 in the course of discovery of a new sporadic simple group HS. Currently only 7 SRGs with no triangles are known, those on 5, 10, 16, 50, 56, 77 and 100 vertices. All them appear as induced subgraphs of the (unique) SRG with no triangles on 100 vertices.

This survey talk is influenced by our recent careful investigation (jointly with A. Woldar and M. Ziv-Av) of two texts by Dale Marsh Mesner (1923-2009) dated by 1956 and 1964. It turns out that the SRG with the parameters (1100, 22, 0, 6) was discovered by Mesner as early as in his thesis (1956), while in 1964 the proof of the uniqueness of the graph, denoted $NL_2(10)$ by Mesner, was presented in his notes. Moreover, in framework of his investigation of the graph $NL_2(10)$ the  elements of a general theory of SRGs with no triangles were developed, with a special emphasis to SRGs of negative Latin square type with no triangles.

The goal of the suggested survey talk is to bring together diverse facts from the theory of SRGs with no triangles
in order to promote fruitful discussions of the participants of the workshop. Special attention will be payed to ideas which stem from Mesner, as well as from N. Biggs, P. Cameron, G. Higman, M. Macaj and J. Siran and other authors.