Construct a compatible auxiliary graph for the first feasible parameter set

Construct a compatible 30-regular auxiliary graph on 54 vertices with spectrum \(\{[30]^1,[6]^9,[3]^8,[-3]^{36}\}\) for the transversal design, equivalently orthogonal array \(\mathrm{OA}(54,4,3,2)\), associated with the first row of Table 1, so that the auxiliary-graph identities required to produce a biregular graph with three distinct eigenvalues are satisfied.

Background

The first row of Table 1 corresponds to a semiregular group divisible design with parameters $(12,54;18,4;0,6;4,3)$. Because $\lambda_1=0$ and $k=m=4$, the design is a transversal design, equivalently an orthogonal array $\mathrm{OA}(54,4,3,2)$. The paper explains that an orthogonal array with these parameters can be constructed by duplicating rows of a linear array over $\mathbb F_3$.

What remains unresolved is not the existence of the orthogonal array itself, but the existence of a compatible auxiliary graph on its 54 blocks. Such a graph must be 30-regular and have spectrum $\{[30]^1,[6]^9,[3]^8,[-3]^{36}\}$, while also satisfying the compatibility conditions that allow the group divisible design and auxiliary graph to yield a graph with three distinct eigenvalues.