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.

中文翻译：

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多块图的距离矩阵：行列式和逆矩阵

如果连接图的每个块都是完整的多部分图，则将其称为多块图。我们考虑具有距离矩阵具有非零辅助因子的块的多块图的距离矩阵。在这种情况下，如果一个多块图的距离矩阵是可逆的，我们会发现该逆矩阵是类拉普拉斯矩阵的倍数的一阶扰动。我们还为一类多块图提供了距离矩阵的逆，其中一个块的距离矩阵的辅助因子为零。