Uniform Spanning Trees of Planar Graphs

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Abstract

Let G be a finite connected graph. A spanning tree T of G is a connected subgraph of G that contains no cycles and such that every vertex of G is incident to at least one edge of T. The set of spanning trees of a given finite connected graph is obviously finite and hence we may draw one uniformly at random. This random tree is called the uniform spanning tree (UST) of G. This model was first studied by Kirchhoff who gave a formula for the number of spanning trees of a given graph and provided a beautiful connection with the theory of electric networks. In particular, he showed that the probability that a given edge {x, y} of G is contained in the UST equals ℛeff(x↔ y; G); we prove this fundamental formula in Sect. 7.2 (see Theorem 7.2).

Original languageEnglish
Title of host publicationLecture Notes in Mathematics
PublisherSpringer Verlag
Pages89-103
Number of pages15
DOIs
StatePublished - 2020

Publication series

NameLecture Notes in Mathematics
Volume2243
ISSN (Print)0075-8434
ISSN (Electronic)1617-9692

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