Shilla graphs with $b = 5$ and $b = 6$

A $Q$-polynomial Shilla graph with ${b = 5}$ has intersection arrays ${\{105t,4(21t+1),16(t+1);}$ ${1,4 (t+1),84t\}}$, $t\in\{3,4,19\}$. The paper proves that distance-regular graphs with these intersection arrays do not exist. Moreover, feasible intersection arrays of $Q$-polynomial Shilla graphs with $b = 6$ are found.