Related papers: Cayley Trees and Bethe Lattices, a concise analysi…

We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 \pi i/3}]$: there are only four such lattices,…

Statistical field theory methods have been very successful with a number of random graph and random matrix problems, but it is challenging to apply these methods to graphs with prescribed degree sequences due to the extensive number of…

I present a brief introduction to the lattice formulation of quantum field theory, and discuss the use of lattice simulations for studies in particle physics phenomenology. The computation of $f_B$, the decay constant of the $B$-meson, is…

We study some sequences of polynomials that appear when we consider the successive derivatives of the tree function (or Lambert's W function). We show in particular that they are related with a generalization of Cayley trees, called Greg…

In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed points. We prove that there…

To most mathematicians and computer scientists the word ``tree'' conjures up, in addition to the usual image, the image of a connected graph with no circuits. In the last few years various types of trees have been the subject of much…

The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two…

In 1993 van den Berg and Kesten proved a strict monotonicity theorem for first passage percolation on $\mathbb{Z}^d$, $d \ge 2$: given two probability measures $\nu$ and $\tilde{\nu}$ with finite mean, if $\tilde{\nu}$ is strictly more…

The aim of this paper is twofold. First, we study eigenvalues and eigenvectors of the adjacency matrix of a bond percolation graph when the base graph is finite and well approximated locally by an infinite regular graph. We relate…

Owing to their interesting spectral properties, the synthetic crystals over lattices other than regular Euclidean lattices, such as hyperbolic and fractal ones, have attracted renewed attention, especially from materials and meta-materials…

Materials science and the study of the electronic properties of solids are a major field of interest in both physics and engineering. The starting point for all such calculations is single-electron, or non-interacting, band structure…

A perfect forest is a spanning forest of a connected graph $G$, all of whose components are induced subgraphs of $G$ and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a…

Tree ensembles (TEs) find a multitude of practical applications. They represent one of the most general and accurate classes of machine learning methods. While they are typically quite concise in representation, their operation remains…

We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers $d\geq 2$ and $m \ge 1$, we consider an uncountable family of groups of automorphisms of the…

Numerical simulations of coupled map lattices (CMLs) and other complex model systems show an enormous phenomenological variety that is difficult to classify and understand. It is therefore desirable to establish analytical tools for…

A covariance graph is an undirected graph associated with a multivariate probability distribution of a given random vector where each vertex represents each of the different components of the random vector and where the absence of an edge…

The lattice field theory approach to the statistical mechanics of a classical Coulomb gas [R. Coalson and A. Duncan, J. Chem. Phys. 97,5653(1992)] is generalized to include charged polymer chains. Saddle-point analysis is done on the…

A Bayesian treatment of latent directed graph structure for non-iid data is provided where each child datum is sampled with a directed conditional dependence on a single unknown parent datum. The latent graph structure is assumed to lie in…

Strong cospectrality is an equivalence relation on the set of vertices of a graph that is of importance in the study of quantum state transfer in graphs. We construct families of abelian Cayley graphs in which the number of mutually…

We prove Leavitt path algebra versions of the two uniqueness theorems of graph C*-algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras and give necessary and sufficient conditions for their…