Spectral characterization of the Hamming graphs

We show that the Hamming graph H(3,q) with diameter three is uniquely determined by its spectrum for q⩾36. Moreover, we show that for given integer D⩾2, any graph cospectral with the Hamming graph H(D,q) is locally the disjoint union of D copies of the complete graph of size q-1, for q large enough.