Graphs $\Gamma$ of diameter 4 for which $\Gamma_{3,4}$ is a strongly regular graph with $\mu=4,6$

We consider antipodal graphs $\Gamma$ of diameter 4 for which $\Gamma_{1,2}$ is a strongly regular graph. A.A. Makhnev and D.V. Paduchikh noticed that, in this case, $\Delta=\Gamma_{3,4}$ is a strongly regular graph without triangles. It is known that in the cases $\mu=\mu(\Delta)\in \{2,4,6\}$ there are infinite series of admissible parameters of strongly regular graphs with $k(\Delta)=\mu(r+1)+r^2$, where $r$ and $s=-(\mu+r)$ are nonprincipal eigenvalues of $\Delta$. This paper studies graphs with $\mu(\Delta)=4$ and 6. In these cases, $\Gamma$ 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.