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Asymptotic enumeration of spanning trees
 Combin. Probab. Comput
, 2005
"... 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 ..."
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Cited by 29 (6 self)
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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 quasitransitive 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
Edge Isoperimetric Problems on Graphs
 Bolyai Math. Series
"... 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 ..."
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Cited by 16 (5 self)
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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...
Evolving sets, mixing and heat kernel bounds
 Probab. Theory Rel. Fields
, 2005
"... 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 boun ..."
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Cited by 10 (2 self)
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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 continuoustime 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
Mixing for Markov Chains and Spin Systems
 LECTURES GIVEN AT THE 2005 PIMS SUMMER SCHOOL IN PROBABILITY HELD AT THE UNIVERSITY OF BRITISH COLUMBIA FROM JUNE 6 THROUGH JUNE 30.
, 2005
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The isoperimetric constant of the random graph process, Random Structures Algorithms 32
"... The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of ∂S S, taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, ˜ G(t), is a sequence of graphs, where G(0) ˜ ..."
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Cited by 4 (3 self)
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The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of ∂S S, taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, ˜ G(t), is a sequence of graphs, where G(0) ˜ is the edgeless graph on n vertices, and G(t) ˜ is the result of adding