Fairest edge usage and minimum expected overlap for random spanning trees

Expected Lengths of Minimum Spanning Trees for Non-identical Edge Distributions

An exact general formula for the expected length of the minimal spanning tree (MST) of a connected (possibly with loops and multiple edges) graph whose edges are assigned lengths according to independent (not necessarily identical) distributed random variables is developed in terms of the multivariate Tutte polynomial (alias Potts model). Our work was inspired by Steele’s formula based on two-v...

On random minimum lenght spanning trees

On Random Minimum Length Spanning Trees

Suppose that we are given a complete graph K n on n vertices together with lengths on the edges which are independent identically distributed non-negative random variables. Suppose that their common distribution function F satisfies F(0) =0, F is differentiable from the right at zero and D = F~. (0)>0. Let X denote a random variable with this distribution. Let L~ denote the (random) length of t...

On the Difference of Expected Lengths of Minimum Spanning Trees

An exact formula for the expected length of the minimum spanning tree of a connected graph, with independent and identical edge distribution, is given, which generalizes Steele’s formula in the uniform case. For a complete graph, the difference of expected lengths between exponential distribution, with rate one, and uniform distribution on the interval (0, 1) is shown to be positive and of rate...

Minimal Spanning Trees for Graphs with Random Edge Lengths

The theory of the minimal spanning tree (MST) of a connected graph whose edges are assigned lengths according to independent identically distributed random variables is developed from two directions. First, it is shown how the Tutte polynomial for a connected graph can be used to provide an exact formula for the length of the minimal spanning tree under the model of uniformly distributed edge l...