Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees

The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems [Discrete Appl. Math. 155 (3) 2006, 374-385; Adv. Appl. Math. 34 (2005), 138-155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80 (1997), 37-56] determined extremal values of σT(w)/σT(u), σT(w)/σT(v), σ(T)/σT(v), and σ(T)/σT(w), where T is a tree on n vertices, v is in the centroid of the tree T, and u,w are leaves in T. In this paper we test how far the negative correlation between distances and subtrees go if we look for the extremal values of FT(w)/FT(u), FT(w)/FT(v), F(T)/FT(v), and F(T)/FT(w), where T is a tree on n vertices, v is in the subtree core of the tree T, and u,w are leaves in T-the complete analogue of [Discrete Appl. Math. 80 (1997), 37-56], changing distances to the number of subtrees. We include a number of open problems, shifting the interest towards the number of subtrees in graphs

Random subtrees of complete graphs

We study the asymptotic behavior of four statistics associated with subtrees of complete graphs: the uniform probability $p_n$ that a random subtree is a spanning tree of $K_n$, the weighted probability $q_n$ (where the probability a subtree is chosen is proportional to the number of edges in the subtree) that a random subtree spans and the two expectations associated with these two probabilities. We find $p_n$ and $q_n$ both approach $e^{-e^{-1}}\approx .692$, while both expectations approach the size of a spanning tree, i.e., a random subtree of $K_n$ has approximately $n-1$ edges

Trickle-down processes and their boundaries

It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in one-by-one at a distinguished source vertex, successive particles proceed along directed edges according to an appropriate stochastic mechanism, and each particle comes to rest once it encounters an unoccupied vertex. Examples include the binary and digital search tree processes, the random recursive tree process and generalizations of it arising from nested instances of Pitman's two-parameter Chinese restaurant process, tree-growth models associated with Mallows' phi model of random permutations and with Schuetzenberger's non-commutative q-binomial theorem, and a construction due to Luczak and Winkler that grows uniform random binary trees in a Markovian manner. We introduce a framework that encompasses such Markov chains, and we characterize their asymptotic behavior by analyzing in detail their Doob-Martin compactifications, Poisson boundaries and tail sigma-fields.Comment: 62 pages, 8 figures, revised to address referee's comment

Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

Consider a low temperature stochastic Ising model in the phase coexistence regime with Markov semigroup $P_t$. A fundamental and still largely open problem is the understanding of the long time behavior of \d_\h P_t when the initial configuration \h is sampled from a highly disordered state $\nu$ (e.g. a product Bernoulli measure or a high temperature Gibbs measure). Exploiting recent progresses in the analysis of the mixing time of Monte Carlo Markov chains for discrete spin models on a regular $b$-ary tree \Tree^b, we tackle the above problem for the Ising and hard core gas (independent sets) models on \Tree^b. If $\nu$ is a biased product Bernoulli law then, under various assumptions on the bias and on the thermodynamic parameters, we prove $\nu$-almost sure weak convergence of \d_\h P_t to an extremal Gibbs measure (pure phase) and show that the limit is approached at least as fast as a stretched exponential of the time $t$. In the context of randomized algorithms and if one considers the Glauber dynamics on a large, finite tree, our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time, provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution.Comment: 35 page

Maximum Agreement Subtrees and H\"older homeomorphisms between Brownian trees

We prove that the size of the largest common subtree between two uniform, independent, leaf-labelled random binary trees of size $n$ is typically less than $n^{1/2-\varepsilon}$ for some $\varepsilon>0$. Our proof relies on the coupling between discrete random trees and the Brownian tree and on a recursive decomposition of the Brownian tree due to Aldous. Along the way, we also show that almost surely, there is no $(1-\varepsilon)$-H\"older homeomorphism between two independent copies of the Brownian tree.Comment: 32 pages, 5 figure

Trees, Partitions, and Other Combinatorial Structures

This dissertation contains work on three main topics. Chapters 1 through 4 provide complexity results for the single cut-or-join model for genome rearrangement. Genomes will be represented by binary strings. Let S be a finite collection of binary strings, each of the same length. Define M to be the collection of medians – binary strings μ which minimize Sigma v belongs to S H(μ,v) where H is the Hamming distance. For any non-negative function f(x), define Z(f(x), S) to be (Sigma μ belongs to M) (Pi v belongs to S)f(H(μ,v)). We study the complexity of calculating Z(f(x), S), with respect to the number of strings in S and their length. If the leaves of a star are labeled with the strings in S, then Z(x!, S) counts the pairs of functions where one selects a median μ for S and the other assigns, to each v belongs to S, a permutation of coordinates in which μ and v differ. This relates to the small parsimony problem for genome rearrangement. We show that it is #P-complete to calculate Z(x!, S) and give similar results for other functions f(x). We also consider an analogous problem when the leaves of a binary tree are labeled. This is joint work with István Miklós. Chapters 5 and 6 explore tree invariants. In particular, Chapter 5 examines the eccentricity of a vertex, eccT (v) = maxu2T dT (v, u) where dT (u, v) is the number of edges along the path connecting u and v in T. This was one of the first, distancebased, tree invariants studied (Jordan 1869). The total eccentricity of a tree, Ecc(T), is the sum of the eccentricities of its vertices. We determine extremal values and characterize extremal tree structures for the ratios Ecc(T)/ eccT (u), Ecc(T)/ eccT (v), eccT (u)/ eccT (v), and eccT (u)/ eccT (w) where u,w are leaves of T and v is in the center of T. Analogous problems have been resolved for other tree invariants including distance (Barefoot, Entringer, and Székely 1997) and the number of subtrees (Székely and Wang 2013). In addition, we determine the tree structures that minimize and maximize total eccentricity among trees with a given degree sequence. This is joint work with László Székely and Hua Wang. Chapter 6 compares three different middle parts of a tree. Different middle parts such as center, centroid, subtree core have been defined and studied. We want to provide some general insights on the difference between them and consider how far apart (with given order of the tree) two different ‘middle point’ can be and when such maximum distances are achieved. This study, after conducted on general trees, is naturally extended to trees with restricted degrees or diameter due to the evident correlation between these restrictions and the maximum distance between middle parts. Some related interesting questions arise that may be of interest independently. This is joint work with László Székely, Hua Wang, and Shuai Yuan. Chapter 7 studies a problem related to Baranyai’s Theorem. This guarantees that whenever k divides n, there is a partition of ([n]Ck) into rows such that each row is itself a partition of [n]. Baranyai (1973) used graph flows to give an existence proof for this 118 year old conjecture. We are interested in the structure of these partitions. For k = 2, there is a circular configuration which yields a straightforward construction. Beth (1974) found an algebraic construction for k = 3. However neither method has a known extension to larger k. We consider a new construction for k = 2 which makes use of a bijection between partitions and labeled trees. It is our hope that this type of connection will lead to a more general construction of Baranyai partitions. This is joint work with László Székely