We study graphs in which for every edge and all -subgraphs are 2-cocliques. We give a description of connected edge-regular graphs for . In particular, the following examples confirm that the inequality is a sharp bound for strong regularity: the -gon, the icosahedron graph, the graph in and the distance-regular graph of diameter 4 with intersection massive , which is an antipodal 3-covering of the strongly regular graph with parameters .