Abstract
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-variable Tutte polynomial under the model of uniformly identically distributed edge lengths. Applications to wheel graphs and cylinder graphs are given under two types of edge distributions.
Highlights
For a finite and connected graph G = (V, E) with vertex set V and edge set E, we denote the total length of its minimum spanning tree (MST) as LMST (G) = ξe, (1) e∈E(M ST (G))where ξe is the length of the edge e ∈ E
We provide an exact formula for the expected lengths of MSTs for any finite, connected graph G, in which the edge length distributions are not necessarily identical
Since it is natural to divide the edges of the cylinder graph into two types, we study LMST (Pn × Ck) under general non-identical edge distribution
Summary
Our proposed general formula applies to non-simple graphs, i.e. graphs with loops (edges joining a vertex to itself) or multiple edges (two or more edges connecting the same pair of vertices) This feature together with the non-i.i.d. edge length assumption enables us to study the random minimum spanning tree problem in much more complicated situations. These are used in the proof of Theorem 1, where the relationship between the standard Tutte polynomial and the expected length of MST is reviewed and analyzed.
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