Related papers: Self-avoiding walks and multiple context-free lang…

We give necessary and sufficient conditions for lobe-transitivity of locally finite and locally countable graphs whose connectivity equals 1. We show further that, given any biconnected graph $\Lambda$ and a "code" assigned to each orbit of…

Let $G$ be a finite abelian group written additively with identity $0$, and $\Omega$ be an inverse closed generating subset of $G$ such that $0\notin \Omega$. We say that $ \Omega $ has the property \lq\lq{}$us$\rq\rq{} (unique summation),…

For each $n,r \geq 0$, let $KG(n,r)$ denote the Kneser Graph; that whose vertices are labeled by $r$-element subsets of $n$, and whose edges indicate that the corresponding subsets are disjoint. Fixing $r$ and allowing $n$ to vary, one…

The goal of this paper is to construct and describe certain arithmetic subgroups of the automorphism group of a partially commutative group. More precisely, given an arbitrary finite graph $\Gamma$ we construct an arithmetic subgroup…

We consider self-avoiding walk on finite graphs with large girth. We study a few aspects of the model originally considered by Lawler, Schramm and Werner on finite balls in Z^d. The expected length of a random self avoiding path is…

Recently researchers working in the LFG framework have proposed algorithms for taking advantage of the implicit context-free components of a unification grammar [Maxwell 96]. This paper clarifies the mathematical foundations of these…

We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of "basic" graphs…

A subshift of finite type over finitely many symbols can be described as a collection of all infinite walks on a digraph with at most a single edge from a vertex to another. The associated finite set $\F$ of forbidden words is a constraint…

We investigate families of infinite automata for context-sensitive languages. An infinite automaton is an infinite labeled graph with two sets of initial and final vertices. Its language is the set of all words labelling a path from an…

We have analysed the recently extended series for the number of self-avoiding walks (SAWs) $C_L(1)$ that cross an $L \times L$ square between diagonally opposed corners. The number of such walks is known to grow as $\lambda_S^{L^2}.$ We…

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional path algebra KQ of a quiver Q. Under certain assumptions on the action of G, we show the existence of a certain kind of…

The \emph{word problem} of a group $G = \langle \Sigma \rangle$ can be defined as the set of formal words in $\Sigma^*$ that represent the identity in $G$. When viewed as formal languages, this gives a strong connection between classes of…

Large language models (LLMs) are increasingly applied to analyzing wearable sensing data, which are long-term, multimodal, and highly personalized. A key challenge is context selection: providing insufficient context limits reasoning, while…

By analogy with the Cayley graph of a group with respect to a finite generating set or the Cayley--Abels graph of a totally disconnected, locally compact group, we detail countable connected graphs associated to Polish groups that we term…

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$ for each $x\neq y$. The set of word-representable graphs generalizes several…

The distinguishing number $\operatorname D(G)$ of a graph $G$ is the least cardinal $d$ such that $G$ has a labeling with $d$ labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite,…

The connective constant $\mu(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We investigate the validity of the inequality $\mu \ge \phi$ for infinite, transitive, simple,…

We study the asymptotic behavior of ``true" self-avoiding random walks on general infinite locally finite trees. In this model, the walk starts at the root and, at each step, from its current vertex chooses a neighboring edge to traverse…

The study of graph queries in database theory has spanned more than three decades, resulting in a multitude of proposals for graph query languages. These languages differ in the mechanisms. We can identify three main families of languages,…

We discuss functions from edges and vertices of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced labelings. The set of balanced labelings forms an Abelian…