On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws

The triangle-free Krein graph Kre$(r)$ is strongly regular with parameters $((r^2+3r)^2,$ $r^3+3r^2+r,0,r^2+r)$. The existence of such graphs is known only for $r=1$ (the complement of the Clebsch graph) and $r=2$ (the Higman–Sims graph). A. L. Gavrilyuk and A. A. Makhnev proved that the graph Kre$(3)$ does not exist. Later Makhnev proved that the graph Kre$(4)$ does not exist. The graph Kre$(r)$ is the only strongly regular triangle-free graph in which the antineighborhood of a vertex Kre$(r)'$ is strongly regular. The graph Kre$(r)'$ has parameters $((r^2+2r-1)(r^2+3r+1),r^3+2r^2,0,r^2)$. This work clarifies Makhnev's result on graphs in which the neighborhoods of vertices are strongly regular graphs without $3$-cocliques. As a consequence, it is proved that the graph Kre$(r)$ exists if and only if the graph Kre$(r)'$ exists and is the complement of the block graph of a quasi-symmetric $2$-design.