Arising from Hoffman and Singleton's study of Moore graphs, strongly regular graphs play an important role in algebraic graph theory. Strongly regular graphs can be construct from geometric objects, such as generalized quadrangles and certain geometric properties, such as having a regular point, can be studied in the context of graphs. We study pseudo-geometric strongly regular graphs whose second subconstituent with respect to a vertex is a cover of a strongly regular graph or a complete graph. By studying the structure of such graphs, we characterize all graphs containing such a vertex, thereby, answering a question posed by Gardiner, Godsil, Hensel, and Royle. As a by-product of our characterisation, we are able to give new constructions of infinite families of strongly regular graphs and compute many small sporadic examples, in particular, we find 135478 new strongly regular graphs with parameters (85,20,3,5) and 27 039 strongly regular graphs with parameters (156, 30, 4, 6).

This is joint work with Edwin van Dam.