A Graph Bottleneck Inequality
نویسنده
چکیده
For a weighted directed multigraph, let fij be the total weight of spanning converging forests that have vertex i in a tree converging to j. We prove that fij fjk = fik fjj if and only if every directed path from i to k contains j (a graph bottleneck equality). Otherwise, fij fjk < fik fjj (a graph bottleneck inequality). In a companion paper [1] (P. Chebotarev, A new family of graph distances, arXiv preprint math.CO/0810.2717, 2008. http://arXiv.org/abs/0810.2717. Submitted), this inequality underlies, by ensuring the triangle inequality, the construction of a new family of graph distances. This stems from the fact that the graph bottleneck inequality is a multiplicative counterpart of the triangle inequality for proximities.
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