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Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : ..."
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Cited by 39 (18 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
A recurrence related to trees
 Proceedings of the American Mathematical Society
, 1989
"... Abstract. The asymptotic behavior of the solutions to an interesting class of recurrence relations, which arise in the study of trees and random graphs, is derived by making uniform estimates on the elements of a basis of the solution space. We also investigate a family of polynomials with integer c ..."
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Cited by 38 (4 self)
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Abstract. The asymptotic behavior of the solutions to an interesting class of recurrence relations, which arise in the study of trees and random graphs, is derived by making uniform estimates on the elements of a basis of the solution space. We also investigate a family of polynomials with integer coefficients, which may be called the "tree polynomials." There are n" ~ (n 1)! sequences of edges between vertices (0.1) ux—vx.un_x—vn_x, \<uk<vk<n, that define a free tree on {1,...,«}, because there are n" ~ free trees on n labeled vertices and every such tree has n 1 edges. If we consider each of these n (n — 1)! sequences to be equally likely, the probability that unX and vn_x belong respectively to components of sizes k and n k based on the first « 2 edges is '^oer^r'■•<*< • ■ Knuth and Schönhage [9, §§912] considered treeconstruction algorithms whose analysis depended on the solution of the recurrence (°3) Xn=Cn+ E Pnk(xk+Xnk) 0<k<n for various sequences (cn). The purpose of the present note is to extend the results of [9] and to consider related sequences of functions whose exact and asymptotic values arise in a variety of algorithms. Much of the analysis below, as in [9], depends on properties of the formal power series tt\A \ TV \ \r^n"~[z " 2, 3 3, 8 4, 125 5, (0.4) T(z) = ^ — — = z + z +z +^z + —z +■■ ■, n>l Received by the editors March 18, 1988.
The Minimal Spanning Tree In A Complete Graph And A Functional Limit Theorem For Trees In A Random Graph.
, 1997
"... . The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tr ..."
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Cited by 23 (3 self)
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. The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tree components of given sizes in a random graph. 1. Introduction and results Assign random weights T ij , 1 i ! j n, to the edges of the complete graph K n with vertex set f1; : : : ; ng, and let W n be the minimum weight of a spanning tree of K n . We assume that the weights are independent and identically distributed, with a uniform distribution on [0; 1]. It was proved by Frieze [5] that W n ! i(3) = 1 X k=1 k \Gamma3 = 1:202 : : : in probability as n ! 1, see also Bollobs [3]. The main purpose of the present paper is to show that W n has an asymptotic normal distribution. Theorem 1. Let W n be the weight of the minimal spanning tree. Then n 1=2 \Gamma W n \Gamma i(3) \Delta ...
Probabilistic bounds on the coefficients of polynomials with only real zeros
 J. Combin. Theory Ser. A
, 1997
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Random planar graphs with n nodes and a fixed number of edges
 Proceedings of the ACMSIAM Symposium on Discrete Algorithms (SODA) 999
, 2005
"... Let P(n,m) be the class of simple labelled planar graphs with n nodes and m edges, and let Rn,q be a graph drawn uniformly at random from P(n, bqnc). We show properties that hold with high probability (w.h.p.) for Rn,q when 1 < q < 3. For example, we show that Rn,q contains w.h.p. linearly man ..."
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Cited by 14 (4 self)
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Let P(n,m) be the class of simple labelled planar graphs with n nodes and m edges, and let Rn,q be a graph drawn uniformly at random from P(n, bqnc). We show properties that hold with high probability (w.h.p.) for Rn,q when 1 < q < 3. For example, we show that Rn,q contains w.h.p. linearly many nodes of each given degree and linearly many node disjoint copies of each given fixed connected planar graph. Additionally, we show that the probability that Rn,q is connected is bounded away from one by a nonzero constant. As a tool we show that (P(n, bqnc)/n!)1/n tends to a limit as n tends to infinity. 1
Random graphs from planar and other addable classes. Topics in Discrete Mathematics
, 2006
"... We study various properties of a random graph Rn, drawn uniformly at random from the class An of all simple graphs on n labelled vertices that satisfy some given property, such as being planar or having treewidth at most k. In particular, we show that if the class A is ‘small ’ and ‘addable’, then ..."
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Cited by 8 (3 self)
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We study various properties of a random graph Rn, drawn uniformly at random from the class An of all simple graphs on n labelled vertices that satisfy some given property, such as being planar or having treewidth at most k. In particular, we show that if the class A is ‘small ’ and ‘addable’, then the probability that Rn is connected is bounded away from 0 and from 1. As well as connectivity we study the appearances of subgraphs, and thus also vertex degrees and the numbers of automorphisms. We see further that if A is ‘smooth ’ then we can make much more precise statements for example concerning connectivity. 1
MSO Zero One Laws on Random Labelled Acyclic Graphs
 Discrete Math
"... Key words: random labelled trees, monadic second order zeroone laws, second order fraisseehrenfeucht games, second moment method PACS: 1 ..."
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Cited by 4 (2 self)
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Key words: random labelled trees, monadic second order zeroone laws, second order fraisseehrenfeucht games, second moment method PACS: 1
Perpendicular dissections of space
"... Abstract. For each pair (Qi, Qj) of reference points and each real number r there is a unique hyperplane h ⊥ QiQj such that d(P, Qi) 2 − d(P, Qj) 2 = r for points P in h. Take n reference points in dspace and for each pair (Qi, Qj) a finite set of real numbers. The corresponding perpendiculars form ..."
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Abstract. For each pair (Qi, Qj) of reference points and each real number r there is a unique hyperplane h ⊥ QiQj such that d(P, Qi) 2 − d(P, Qj) 2 = r for points P in h. Take n reference points in dspace and for each pair (Qi, Qj) a finite set of real numbers. The corresponding perpendiculars form an arrangement of hyperplanes. We explore the structure of the semilattice of intersections of the hyperplanes for generic reference points. The main theorem is that there is a real, additive gain graph (this is a graph with an additive real number associated invertibly to each edge) whose set of balanced flats has the same structure as the intersection semilattice. We examine the requirements for genericity, which are related to behavior at infinity but remain mysterious; also, variations in the construction rules for perpendiculars. We investigate several particular arrangements with a view to finding the exact numbers of faces of each dimension. The prototype, the arrangement of all perpendicular bisectors, was studied by Good and Tideman, motivated by a geometric voting theory. Most of our particular examples are suggested by extensions of that theory in which voters exercise finer discrimination. Throughout, we propose many research problems. Postpublication revision 8 May 2002: added reference to Voronoi in §9: “that goes back to the original paper [25a]”. 1.
AN OPERATOR APPROACH TO THE PRINCIPLE OF INCLUSION AND EXCLUSION
"... ABSTRACT. Using an operator approach we derive SylvesterWhitworth formulae for sets A'a. By the same token we treat the problem where both sets of A's and B's are involved. Our result extends the SylvesterWhitworth inclusion and exclusion formula to the resolution of the number of e ..."
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ABSTRACT. Using an operator approach we derive SylvesterWhitworth formulae for sets A'a. By the same token we treat the problem where both sets of A's and B's are involved. Our result extends the SylvesterWhitworth inclusion and exclusion formula to the resolution of the number of elements in exactly mi sets of A's and mi sets of B's respectively. The formula are applied to the complete graph and complete bipartite graph. The enumeration of spanning subgraphs with any preassigned number of disconnected cycles is solved, together with the case where any preassigned number of vertices have degree one.
MSO Zero One Laws on Random Acyclic Graphs
, 1999
"... We use Ehrenfeuchttype games to prove that Monadic Second Order logic admits labelled and unlabelled zeroone laws for random free trees, generating the same complete almost sure theory. Our method will be to dissect random trees to get a picture of what almost all random free trees look like. ..."
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We use Ehrenfeuchttype games to prove that Monadic Second Order logic admits labelled and unlabelled zeroone laws for random free trees, generating the same complete almost sure theory. Our method will be to dissect random trees to get a picture of what almost all random free trees look like. In particular, we will prove that for any fixed rooted tree T, almost every sufficiently large free tree has a subtree isomorphic to T. This research was partially supported by NSF grant CCR 9403463. I would also like to thank USF for time off for the research on this project. 1 Contents 1 Introduction 3 2 Some Logic 3 2.1 Monadic Second Order Logic . . . . . . . . . . . . . . . . . . . 4 2.1.1 Constructing MSO . . . . . . . . . . . . . . . . . . . . 4 2.1.2 MSO Bisimulation Games . . . . . . . . . . . . . . . . 5 2.2 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . 9 3 Probability on Labelled Tree...