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A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Recently, Csikvari proved the existence of integral trees of any even diameter. In the odd case, int...
Sub-Gaussian tail bounds for the width and height of conditioned Galton--Watson trees
Sub-Gaussian tail conditioned Galton--Watson trees
2010/11/23
We study the height and width of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes. Under this conditioning, we deriv...
Central limit theorem for biased random walk on multi-type Galton-Watson trees
Central limit theorem random walk multi-type Galton-Watson trees
2010/11/23
Let T be a rooted multi-type Galton-Watson (MGW) tree of finitely many types with at least one offspring at each vertex, and an offspring distribution with exponential tails. The lambda-biased random...
Fires on trees
Fires on trees math
2010/11/15
We consider random dynamics on the edges of a uniform Cayley tree with $n$ vertices, in which edges are either inflammable, fireproof, or burt. Every inflammable edge is replaced by a fireproof edge a...
On the conjugacy problem for finite-state automorphisms of regular rooted trees
finite-state automorphisms regular rooted trees math
2010/11/15
We study the conjugacy problem in the automorphism group $Aut(T)$ of a regular rooted tree $T$ and in its subgroup $FAut(T)$ of finite-state automorphisms. We show that under the contracting conditio...
Spectral characterization of a specific class of trees
Spectral characterization specific class of trees
2010/11/9
In this paper, it is shown that the graph $T_4(p,q,r)$ is determined by its Laplacian spectrum and there are no two non-isomorphic such graphs which are cospectral with respect to adjacency spectrum. ...
Lattice Polynomials, 12312-Avoiding Partial Matchings and Even Trees
Lattice Polynomials Partial Matchings Even Trees
2010/11/22
The lattice polynomials $L_{i,j}(x)$ are introduced by Hough and Shapiro as a weighted count of certain lattice paths from the origin to the point $(i,j)$. In particular, $L_{2n, n}(x)$ reduces to th...
On the solution of a quadratic vector equation arising in Markovian Binary Trees
quadratic vector equation arising Markovian Binary Trees
2010/11/11
We present some advances, both from a theoretical and from a computational point of view, on a quadratic vector equation (QVE) arising in Markovian Binary Trees. Concerning the theoretical advances, ...
Star graphs: threaded distance trees and E-sets
Star graphs threaded distance trees E-sets
2010/11/22
The distribution of distances in the star graph $ST_n$, ($1established, and subsequently a threaded binary tree is obtained that realizes an orientation of $ST_n$ whose levels are given...
We consider the Neumann Sturm-Liouville problem de ned on trees such that the ratios of lengths of edges are not necessarily rational. It is shown that the potential function of the Sturm-Liouville op...
Rotor walks on general trees
rotor walk rotor-router infinite tree quasi-random branching process
2010/12/13
The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on
successive ...
Gibbs-non-Gibbs properties for evolving Ising models on trees
non-Gibbsianness Ising models tree graphs Glauber dynamics
2010/12/7
In this paper we study homogeneous Gibbs measures on a Cayley tree,subjected to an in nite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gi...
Large deviation results for Multitype Galton-Watson trees
Tree-indexed Markov chain Tree-indexed process random tree
2010/12/7
Using the notion of consistency of empirical measures, under fairly general assumption we
prove a joint large deviation principles in n for the empirical pair measure and empirical
offspring measure...
Compactness properties of weighted summation operators on trees - the critical case
Metrics on trees operators on trees weighted summation operators
2010/12/6
The aim of this paper is to provide upper bounds for the entropy numbers of summation
operators on trees in a critical case. In a recent paper [10] we elaborated a framework of
weighted summation op...
An upper bound for the Hosoya index of trees
Hosoya index of graphs tree eigenvalue of graphs
2010/9/13
The Hosoya index of a graph G is defined as the sum of all the numbers of k - matchings (k ≥ 0) in G. An upper bound for the Hosoya index of trees is presented in this note.