GRAPHS \(Γ\) OF DIAMETER 4 FOR WHICH \(Γ3,4\) IS A STRONGLY REGULAR GRAPH WITH \(μ=4,6\)

We consider antipodal graphs \(Γ\) of diameter 4 for which \(Γ1,2\) is a strongly regular graph. A.A. Makhnev and D.V. Paduchikh noticed that, in this case, \(Δ=Γ3,4\) is a strongly regular graph without triangles. It is known that in the cases \(μ=μ(Δ)∈ \{2,4,6\}\) there are infinite series of admissible parameters of strongly regular graphs with \(k(Δ)=μ(r+1)+r^2\), where \(r\) and \(s=-(μ+r)\) are nonprincipal eigenvalues of \(Δ\). This paper studies graphs with \(μ(Δ)=4\) and 6. In these cases, \(Γ\) has intersection arrays \(\{{r^2+4r+3},{r^2+4r},4,1;1,4,r^2+4r,r^2+4r+3\}\) and \(\{r^2+6r+5,r^2+6r,6,1;1,6,r^2+6r,r^2+6r+5\}\), respectively. It is proved that graphs with such intersection arrays do not exist.