$L_{\infty}$ norm minimization for nowhere-zero integer eigenvectors of the block graphs of Steiner triple systems and Johnson graphs

We study nowhere-zero integer eigenvectors of the block graphs of Steiner triple systems and the Johnson graphs. For the first eigenvalue we obtain the minimums of the $L_{\infty}$ norm for several infinite series of Johnson graphs, including $J(n,3)$ for all $n\geq 63$, as well as general upper and lower bounds. The minimization of the $L_{\infty}$ norm for nowhere-zero integer eigenvectors with the second eigenvalue of the block graph of a Steiner triple system $S$ is equivalent to finding the minimum nowhere-zero flow for Steiner triple system $S$. For the all Assmuss-Mattson Steiner triple systems of the orders greater or equal to $99$ we prove that the minimum flow is bounded above by $5$.