On distance matrices of graphs

Distance well-defined graphs consist of connected undirected graphs, strongly connected directed graphs and strongly connected mixed graphs. Let $G$ be a distance well-defined graph, and let ${\sf D}(G)$ be the distance matrix of $G$. Graham, Hoffman and Hosoya [3] showed a very attractive theorem, expressing the determinant of ${\sf D}(G)$ explicitly as a function of blocks of $G$. In this paper, we study the inverse of ${\sf D}(G)$ and get an analogous theory, expressing the inverse of ${\sf D}(G)$ through the inverses of distance matrices of blocks of $G$ (see Theorem 3.3) by the theory of Laplacian expressible matrices which was first defined by the first author [9]. A weighted cactoid digraph is a strongly connected directed graph whose blocks are weighted directed cycles. As an application of above theory, we give the determinant and the inverse of the distance matrix of a weighted cactoid digraph, which imply Graham and Pollak's formula and the inverse of the distance matrix of a tree.