Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call “tree entropy”, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasi-transitive amenable graphs, extending a result of Burton and Pemantle (1993). §1. Introduction. Methods of enumeration of spanning trees in a finite graph G and relations to various areas of mathematics and physics have been investigated for more than 150 years. The number of spanning trees is often called the complexity of the graph, denoted here by τ(G). The best known formula for the complexity, proved in every basic text on graph

We survey results on edge isoperimetric problems on graphs, present some new results and show some applications of such problems in combinatorics and computer science. 1 Introduction Let G = (V G ; EG ) be a simple connected graph. For a subset A ` VG denote I G (A) = f(u; v) 2 EG j u; v 2 Ag; ` G (A) = f(u; v) 2 EG j u 2 A; v 62 Ag: We omit the subscript G if the graph is uniquely defined by the context. By edge isoperimetric problems we mean the problem of estimation of the maximum and minimum of the functions I and ` respectively, taken over all subsets of VG of the same cardinality. The subsets on which the extremal values of I (or `) are attained are called isoperimetric subsets. These problems are discrete analogies of some continuous problems, many of which can be found in the book of P'olya and Szego [99] devoted to continuous isoperimetric inequalities and their numerous applications. Although the continuous isoperimetric problems have a history of thousand years, the dis...

We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovász and Kannan, can be refined to apply to the maximum relative deviation |p n (x, y)/π(y) − 1 | of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities. 1

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2\Delta graphs, where eG(0) is the edgeless graph on n vertices, and eG(t) is the result of addingan edge to e G(t- 1), uniformly distributed over all the missing edges. We show that in almost every graph process i ( eG(t)) equals the minimal degree of eG(t) as long as the minimal degree is o(log n). Furthermore, we show that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically \Theta (log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to 12, its final value. 1 Introduction Let G = (V, E) be a graph. For each subset of its vertices, S ` V, we define its edge boundary, @S, as the set of all edges with exactly one endpoint in S: