Some extremal families of edge-regular graphs

We build on a previous result concerning regular simple graphs for which there is some λ>0 such that any two adjacent vertices have exactly λ common neighbors, and the union of their neighborhoods includes all but μ=2 vertices. We are concerned here with graphs satisfying such requirements with μ>2. We give an upper bound on the order of the graph in terms of λ and μ. This bound turns out to be sharp if μ/2 divides λ. Further, for such restricted values, the extremal graphs for our inequality turn out to be unique for λ sufficiently large.