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 Series. A, 205:105872, 2024] it was shown that the 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 that paper. The main goal of this article is to fill this gap by showing that the graphs arising from the Hasse diagrams of the Hamming posets are $Q$-polynomial.