On the Q-polynomial property of the full bipartite graph of a Hamming graph

The Q-polynomial property is an algebraic property of distance-regular graphs, that was introduced by Delsarte in his study of coding theory. Many distance-regular graphs admit the Q-polynomial property. Only recently the Q-polynomial property has been generalized to graphs that are not necessarily distance-regular. In J. Combin. Theory Ser. A, 205: 105872, 2024, it was shown that graphs arising from the Hasse diagrams of the so-called attenuated space posets are Q-polynomial. These posets could be viewed as q-analogs of the Hamming posets, which were not studied in J. Combin. Theory Ser. A, 205: 105872, 2024. The main goal of this paper is to fill this gap by showing that the graphs arising from the Hasse diagrams of the Hamming posets are Q-polynomial.