Distance-regular graphs with exactly one positive $q$-distance eigenvalue

In this paper, we study the $q$-distance matrix for a distance-regular graph and show that the $q$-distance matrix of a distance-regular graph with classical parameters ($D,q,\alpha ,\beta$) has exactly three distinct eigenvalues, of which one is zero. Moreover, we study distance-regular graphs whose $q$-distance matrix has exactly one positive eigenvalue.