Distance matrix of a multi-block graph: determinant and inverse

A connected graph is called a multi-block graph if each of its blocks is a complete multipartite graph. We consider the distance matrix of multi-block graphs with blocks whose distance matrices have nonzero cofactor. In this case, if the distance matrix of a multi-block graph is invertible, we find the inverse as a rank one perturbation of a multiple of a Laplacian-like matrix. We also provide the inverse of the distance matrix for a class of multi-block graphs in which the distance matrix of one of its block has zero cofactor.