Deza graphs: A generalization of strongly regular graph

We consider the following generalization of strongly regular graphs. A graph G is a Deza graph if it is regular and the number of common neighbors of two distinct vertices takes on one of two values (not necessarily depending on the adjacency of the two vertices). We introduce several ways to construct Deza graphs, and develop some basic theory. We also list all diameter two Deza graphs which are not strongly regular and have at most 13 vertices.