Abstract
Metric tasks often arise as a simplification of complex and practically important problems on graphs. The correspondence between the search algorithms of the usual shortest paths and Markov chains is shown. From this starting point a sequence of matrix descriptions of undirected graphs is established. The sequence ends with the description of the explicit form of the Moore-Penrose pseudo inversed incidence matrix. Such a matrix is a powerful analytical and computational tool for working with edge flows with conditionally minimal Euclidian norms. The metrics of a graph are represented as its characteristics generated by the norms of linear spaces of edge and vertex flows. The Euclidian metric demonstrates the advantages of the practice of solving problems on graphs in comparison with traditional metrics based on the shortest paths or minimal cuts.
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