On $Q$-polynomial Shilla graphs with $b=6$

Distance-regular graph $\Gamma$ of diameter $3$, having the second eigenvalue $\theta_1= a_3$ is called Shilla graph. For such graph $a=a_3$ devides $k$ and we set $b = b(\Gamma) = k/a$. Further $a_1 = a - b$ and $\Gamma$ has intersection array $\{ab,(a + 1)(b - 1), b_2; 1, c_2, a(b - 1)\}$. I. N. Belousov and A. A. Makhnev found feasible arrays of $Q$-polynomial Shilla graphs with $b=6$: $\{42t,5(7t+1),3(t+3);1,3(t+3),35t\}$, where $t\in \{7,12,17,27,57\}$, $\{312,265,48;1,24,260\}$, $\{372,315,75;1,15,310\}$, $\{624,525,80;1,40,520\}$, $\{744,625,125;1,25,620\}$, $\{930,780,150;1,30,775\}$, $\{1794,1500,200;1,100,1495\}$ or $\{5694, 4750,600;1,300,4745\}$. It is proved in the paper that graphs with intersection arrays $\{372,315,75;1,15,310\}$, $\{744,625,125;1,25,620\}$ and $\{1794,1500,200;1,100,1495\}$ do not exist.