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

Abstract

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 \((r)^{\prime}\) is strongly regular. The graph Kre \((r)^{\prime}\) 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 \((r)^{\prime}\) exists and is the complement of the block graph of a quasi-symmetric \(2\) -design.