Related papers: Couplings of Uniform Spanniing Forests
In their study of fundamental groups of one-dimensional path-connected compact metric spaces, Cannon and Conner have asked: Is there a tree-like object that might be considered the topological Cayley graph? We answer this question in the…
Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and structure of connecting trajectories of the underlying dynamical…
We continue to study Cayley configuration spaces of 1-dof linkages in 2D begun in Part I of this paper, i.e. the set of attainable lengths for a non-edge. In Part II, we focus on the algebraic complexity of describing endpoints of the…
We define two families of determinantal random spanning subgraphs of a finite connected graph, one supported by acyclic spanning subgraphs (spanning forests) with fixed number of connected components, the other by connected spanning…
We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability…
A bi-Cayley graph over the cyclic group $(\mathbb{Z}_n, +)$ is called a bicirculant graph. Let $\Gamma=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=-R\subseteq \mathbb{Z}_n\setminus \{0\}$ and $T={-}T\subseteq…
In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually…
We prove a sharp structural result concerning finite colorings of pairs in well-founded trees.
Working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree-like structures to a given graph. We find a correspondence between the solutions of random K-satisfiability problem and those of spanning…
We start by introducing avoidance coupling of Markov chains, with an overview of existing results. We then introduce and motivate a new notion, uniform avoidance coupling. We show that the only Markovian avoidance coupling on a cycle is of…
Binomial Cayley graphs are obtained by considering the binomial coefficient of the weight function of a given Cayley graph and a natural number. We introduce these objects and study two families: one associated with symmetric groups and the…
Let Gamma be a connected, locally finite graph of finite tree width and G be a group acting on it with finitely many orbits and finite node stabilizers. We provide an elementary and direct construction of a tree T on which G acts with…
P\'olya trees are rooted, unlabeled trees on $n$ vertices. This paper gives an efficient, new way to generate P\'olya trees. This allows comparing typical unlabeled and labeled tree statistics and comparing asymptotic theorems with…
Due to their elegant and simple nature, unitary Cayley graphs have been an active research topic in the literature. These graphs are naturally connected to several branches of mathematics, including number theory, finite algebra,…
Neuronal morphology is an essential element for brain activity and function. We take advantage of current availability of brain-wide neuron digital reconstructions of the Pyramidal cells from a mouse brain, and analyze several emergent…
We study asymptotically non-free gauge theories and search for renormalization group invariant (i.e. technically natural) relations among the couplings which lead to successful gauge-Yukawa unification. To be definite, we consider a…
We study Cayley configuration spaces of a class of 1 degree-of-freedom linkages (graphs with specified edge lengths), obtained by dropping an edge from a tree-decomposable graph. The class includes well-known mechanisms based on the…
The connective constant $\mu(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the…
Let $G$ be a (finite or infinite) group, and let $K_G = \mathrm{Cay} ( G;G \smallsetminus \{1\} )$ be the complete graph with vertex set $G$, considered as a Cayley graph of $G$. Being a Cayley graph, it has a natural edge-colouring by sets…
This paper will show when a rooted path tree of a finite directed rooted graph has only finitely many orbits under the action of its undirected automorphism group (i.e. when it is cocompact). This will allow us to specify which trees are…