Three infinite families of Shilla graphs do not exist

A distance-regular graph of diameter 3 with the second eigenvalue $\theta_1=a_3$ is called a Shilla graph. For a Shilla graph $\Gamma$, the number $a=a^3$ divides $k$ and we set $b=b(\Gamma)=k/a$. Three infinite families of Shilla graphs with the following admissible intersection arrays were found earlier: $\{b(b^2-1),b^2(b-1),b^2;1,1,(b^2-1)(b-1)\}$ (I.N. Belousov), $\{b^2(b-1)/2,(b-1)(b^2-b+2)/2,b(b-1)4;1,b(b-1)/4,b(b-1)^2/2\}$ (Koolen, Park), and $\{(s+1)(s^3-1),s^4,s^3;1,s^2,s(s^3-1)\}$. In this paper, it is proved that, in the first family, there exists a unique graph, namely, a generalized hexagon of order 2, whereas there are no graphs in the second or third families.