A q-analogue of distance matrix of bi-block graphs

A q-analogue of the distance matrix, referred to as the q-distance matrix, is obtained from the distance matrix by replacing each nonzero entry $α$ with the sum 1+q+⋯+qα-1. This notion was introduced independently by Bapat, Lal, and Pati~{Ba-Lal-Pati}, and by Yan and Yeh~{Yan}. A connected graph is called a bi-block graph if each of its blocks is a complete bipartite graph. In this paper, we derive explicit formulas for the determinant and the inverse of the q-distance matrix of bi-block graphs. These results both generalize the corresponding formulas for the distance matrix of bi-block graphs obtained in~{Hou3} and extend the results for block graphs in~{Xing} to the class of bi-block graphs.