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Exact Expectations of Minimal Spanning Trees for Graphs With Random Edge Weights

机译:具有随机边缘权重的图的最小生成树的精确期望

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摘要

Two methods are used to compute the expected value of the length of the minimal spanning tree (MST) of a graph whose edges are assigned lengths which are independent and uniformly distributed. The first method yields an exact formula in terms of the Tutte polynomial. As an illustration, the expected length of the MST of the Petersen graph is found to be 34877/12012 = 2.9035 .... A second, more elementary, method for computing the expected length of the MST is then derived by conditioning on the length of the shortest edge. Both methods in principle apply to any finite graph. To illustrate the method we compute the expected lengths of the MSTs for complete graphs.
机译:使用两种方法来计算图的最小生成树(MST)长度的期望值,该图的边被分配了独立且均匀分布的长度。第一种方法根据Tutte多项式得出精确的公式。作为说明,发现彼得森图的MST的预期长度为34877/12012 = 2.9035...。然后,通过对长度进行条件化,得出第二种更基本的方法,用于计算MST的预期长度。最短的边缘。两种方法原则上都适用于任何有限图。为了说明该方法,我们计算了完整图的MST的预期长度。

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