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We provide an algorithm with constant running time that given a weighted tournament T , distinguishes with high probability of success between the cases that T can be represented by a Bradley–Terry model, or cannot even be approximated by one. The same algorithm tests whether the corresponding Markov chain is reversible. [ABSTRACT FROM AUTHOR]

We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of ‘relative compactness’ for such graphs and study sufficient and necessary conditions for this property in terms of various metrics. We then equip graphs satisfying this property with a finite measure and investigate the associated Laplacian and its semigroup. In this context, our results include the trace class property for the semigroup, uniqueness and existence of solutions to the Dirichlet Problem with boundary arising from the natural compactification, an explicit description of the domain of the Dirichlet Laplacian, convergence of the heat semigroup for large times as well as stochastic incompleteness and transience of the corresponding random walk in continuous time. [ABSTRACT FROM AUTHOR]

Abstract: Motivated by work of Diestel and Kühn on the cycle spaces of infinite graphs we study the ramifications of allowing infinite sums in a module . We show that every generating set in this setup contains a basis if the ground set M is countable, but not necessarily otherwise. Given a family , we determine when the infinite-sum span of is closed under infinite sums, i.e.when . We prove that this is the case if R is a field or a finite ring and each element of M lies in the support of only finitely many elements of . This is, in a sense, best possible. We finally relate closures under infinite sums to topological closures in the product space . [Copyright &y& Elsevier]

A well-known result of Benjamini, Lyons, Peres, and Schramm states that if G is a finitely generated Cayley graph of a group Γ , then Γ is amenable if and only if G admits a Γ -invariant random spanning tree with at most two ends. We show that this is equivalent to the existence of a Γ -invariant random spanning double ray in a power of G. [ABSTRACT FROM AUTHOR]