Abstract: Graham, Hoffman and Hosoya (GHH) proved an attractive theorem about the determinant of the distance matrix $D$ of a digraph $G$ as a function of the determinant and the cofactor-sum of the distance matrix of its blocks.

We show two analogs of this result. The first is the $q$-analog of $D$ for digraphs, where we replace all positive entries $i$ by $[i]_q = 1+q+\cdots+q^{i-1}$. A result similar to the GHH theorem is derived. Setting $q=-1$ gives the 0-1 parity matrix of the distance matrix of $G$. When the 0-1 matrix is viewed as an adjacency matrix, some results are obtained about the determinant of the adjacency matrix of graphs. Similar results where each entry $i$ of $D$ is replaced by $[2i]_q$, or more generally $[k i]_q$ are also given.

Another analog is the skew-symmetric version of the distance matrix. In this version, (for graphs only) we fix an ordering $pi$ of the vertices of $G$ for the rows and columns of $D$. All entries of $D$ above the main diagonal are considered positive and all entries below are negative. With this sign pattern, for a specific family of $pi$, we give a GHH type result.