A Neumaier graph is an edge-regular graph with a regular clique. A lot of strongly regular graphs (but clearly not all of them) are indeed Neumaier, but in [3] it was asked whether there are Neumaier graphs that are not strongly regular. This question was only solved recently (see [2]), so now we know there are so-called strictly Neumaier graphs. In this talk I will discuss several new results on (strictly) Neumaier graphs, including (non)-existence results. I will focus on a new construction producing an infinite number of strictly Neumaier graphs, described in [1]. The proofs rely on several results from number theory. I will also discuss a few directions for future research about Neumaier graphs.

References

[1] A. Abiad and W. Castryck and M. De Boeck and J.H. Koolen and S. Zeijlemaker, An infinite class of Neumaier graphs and non-existence results, arXiv:2109.14281 (2021), 22 pp.
[2] G.R. Greaves and J.H. Koolen, Edge-regular graphs with regular cliques, European J. Combin. 71(2018), 194–201.
[3] A. Neumaier, Regular cliques in graphs and special 1 1/2-designs, Finite Geometries and Designs, London Math. Soc. Lecture Note Series vol. 49 (1981), 244–25