Inverse Formulae for $q$-analogues of Bipartite Distance Matrix

We consider two distinct $q$-analogues of the bipartite distance matrix, namely the $q$-bipartite distance matrix and the exponential distance matrix. We provide formulae of the inverse for these matrices, which extend the existing results for the bipartite distance matrix. These investigations lead us to introduce a $q$-analogue version of the bipartite Laplacian matrix.