Appears in collection : Some topics in spectral graph theory

An edge-regular graph is a regular graph such that each edge lies in a constant number of triangles. A regular clique in a graph is a clique such each vertex outside the clique has the same number of neighbours in the clique. In the 1970's Neumaier asked whether there exists edge-regular graph with a regular clique. We call an edge-regular graph a Neumaier graph if it has a regular clique. In this lecture, I will give recent progress on Neumaier graphs.