On distance matrices of graphs
| Autor: | Zhou, Hui, Ding, Qi, Jia, Ruiling |
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| Rok vydání: | 2017 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 21 pages |
| Databáze: | arXiv |
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