So we know that two DRGs with the same intersection array must be co-spectral on their adjacency matrices, i.e. their adjacency matrices have the same set of eigenvalues.
But is this necessarily true also for Laplacian matrices? Over here, the Laplacian matrix is defined as $L(G)=D(G)-A(G)$, where $D(G)$ and $A(G)$ represent the degree matrix and adjacency matrix, respectively.
Other than the Laplacian, what about the other matrices, such as the signless Laplacian matrix $|{L}|(G)=D(G)+A(G)$?
Well, distance-regular graphs with the same intersection array are in particular $k$-regular graphs with the same degree $k$. So for each graph, the Laplacian matrix $L(G)$ is just $kI - A(G)$. If the adjacency matrix has eigenvalues $\lambda_1, \dots, \lambda_n$, then the Laplacian matrix has eigenvalues $k - \lambda_1, \dots, k - \lambda_n$.
Therefore, the two graphs must also be co-spectral with respect to their Laplacian matrices, and (by the same argument) with respect to their signless Laplacian matrices.