Infinite Graphs Research Articles

The profinite completion of the fundamental group of infinite graphs of groups

Given an infinite graph of profinite groups (\({\cal G}\), Γ) we construct a profinite graph of groups (\(\overline {\cal G} ,\overline \Gamma \)) such that Γ is densely embedded in \(\overline \Gamma \), the fundamental profinite group \({\Pi _1}\left( {\overline {\cal G} ,\overline \Gamma } \right)\) is the profinite completion of \({\pi _1}\left( {{\cal G},\Gamma } \right)\) and the standard tree \(S\left( {{\cal G},\Gamma } \right)\) embeds densely in the standard profinite tree \(S\left( {\overline {\cal G} ,\overline \Gamma } \right)\). This answers a Ribes’ question [5, Question 6.7.1]. Generalizing the main results of [8] and [2] we answer two other questions of Ribes [5, Questions 15.11.10 and 15.11.11] proving that a virtually free group G is subgroup conjugacy separable and the normalizer NG(H) of a finitely generated subgroup H of G is dense in \({N_{\hat G}}\left( {\overline H } \right)\). We also give an entirely new description of the fundamental group of a profinite graph of groups using the language of paths.

Classifying Leavitt path algebras up to involution preserving homotopy

We prove that the Bowen–Franks group classifies the Leavitt path algebras of purely infinite simple finite graphs over a regular supercoherent commutative ring with involution where 2 is invertible, equipped with their standard involutions, up to matricial stabilization and involution preserving homotopy equivalence. We also consider a twisting of the standard involution on Leavitt path algebras and obtain partial results in the same direction for purely infinite simple graphs. Our tools are K-theoretic, and we prove several results about (Hermitian, bivariant) K-theory of Leavitt path algebras, such as Poincaré duality, which are of independent interest.

Properties of graphs specified by a regular language

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property varPhi . What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there is a graph satisfying varPhi in the family? We approach this question by using formal languages for specifying families of graphs, in particular by regular sets of words. We show that certain graph properties can be decided by studying the syntactic monoid of the specification language L if a certain torsion condition is satisfied. This condition holds trivially if L is regular. More specifically, we use a natural binary encoding of finite graphs over a binary alphabet varSigma , and we define a regular set mathbb {G}subseteq varSigma ^* such that every nonempty word win mathbb {G} defines a finite and nonempty graph. Also, graph properties can then be syntactically defined as languages over varSigma . Then, we ask whether the automaton mathcal {A} specifies some graph satisfying a certain property varPhi . Our structural results show that we can answer this question for all “typical” graph properties. In order to show our results, we split L into a finite union of subsets and every subset of this union defines in a natural way a single finite graph F where some edges and vertices are marked. The marked graph in turn defines an infinite graph F^infty and therefore the family of finite subgraphs of F^infty where F appears as an induced subgraph. This yields a geometric description of all graphs specified by L based on splitting L into finitely many pieces; then using the notion of graph retraction, we obtain an easily understandable description of the graphs in each piece.

Constructible graphs and pursuit

A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for infinite graphs the situation is not well understood.One of our aims in this paper is to give a graph that is cop-win but not constructible. This is the first known such example. We also show that every countable ordinal arises as the rank of some constructible graph, answering a question of Evron, Solomon and Stahl. In addition, we give a finite constructible graph for which there is no construction order whose associated domination map is a homomorphism, answering a question of Chastand, Laviolette and Polat.Lehner showed that every constructible graph is a weak cop win (meaning that the cop can eventually force the robber out of any finite set). Our other main aim is to investigate how this notion relates to the notion of ‘locally constructible’ (every finite graph is contained in a finite constructible subgraph). We show that, under mild extra conditions, every locally constructible graph is a weak cop win. But we also give an example to show that, in general, a locally constructible graph need not be a weak cop win. Surprisingly, this graph may even be chosen to be locally finite. We also give some open problems.

A note on Neumann problems on graphs

We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations.

Perfect Sampling in Infinite Spin Systems Via Strong Spatial Mixing

We present a simple algorithm that perfectly samples configurations from the unique Gibbs measure of a spin system on a potentially infinite graph $G$. The sampling algorithm assumes strong spatial mixing together with subexponential growth of $G$. It produces a finite window onto a perfect sample from the Gibbs distribution. The run-time is linear in the size of the window, in particular it is constant for each vertex.

Ahlfors regular conformal dimension of metrics on infinite graphs and spectral dimension of the associated random walks

Quasisymmetry is a well-studied property of homeomorphisms between metric spaces, and the Ahlfors regular conformal dimension is a quasisymmetric invariant. In the present paper, we consider the Ahlfors regular conformal dimension of metrics on infinite graphs, and show that this notion coincides with the critical exponent of p -energies. Moreover, we give a relation between the Ahlfors regular conformal dimension and the spectral dimension of a graph.

Generalized entropy for random walks in regular networks and graphs as a super diffusion

In this paper, the entropy of the stochastic processes created by the movement of a walker in a graph is investigated. The Shannon-Khinchin entropy has four axioms that ignore one of them can make the generalized entropy. Here, we investigate the number of different finite paths asymptotically, for determining a generalized entropy. Then, we will study the regular infinite networks and graphs with finite nodes, with two different types of motion.

The generalized porous medium equation on graphs: existence and uniqueness of solutions with $$\ell ^1$$ data

We study solutions of the generalized porous medium equation on infinite graphs. For nonnegative or nonpositive integrable data, we prove the existence and uniqueness of mild solutions on any graph. For changing sign integrable data, we show existence and uniqueness under extra assumptions such as local finiteness or a uniform lower bound on the node measure.

Topological ubiquity of trees

Let ⊲ be a relation between graphs. We say a graph G is ⊲-ubiquitous if whenever Γ is a graph with nG⊲Γ for all n∈N, then one also has ℵ0G⊲Γ, where αG is the disjoint union of α many copies of G.The Ubiquity Conjecture of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation.In this paper we show that all trees are ubiquitous with respect to the topological minor relation, irrespective of their cardinality. This answers a question of Andreae from 1979.

Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs

We study solution sensitivity for nonlinear programs (NLPs) whose structures are induced by graphs. These NLPs arise in many applications such as dynamic optimization, stochastic optimization, optimization with partial differential equations, and network optimization. We show that for a given pair of nodes, the sensitivity of the primal-dual solution at one node against a data perturbation at the other node decays exponentially with respect to the distance between these two nodes on the graph. In other words, the solution sensitivity decays as one moves away from the perturbation point. This result, which we call exponential decay of sensitivity, holds under the strong second-order sufficiency condition and the linear independence constraint qualification. We also present conditions under which the decay rate remains uniformly bounded; this allows us to characterize the sensitivity behavior of NLPs defined over subgraphs of infinite graphs. The theoretical developments are illustrated with numerical examples.

On infinite circulant-balanced complete multipartite graphs decompositions based on generalized algorithmic approaches

Graph theory is a powerful and essential tool for applied scientists and engineers in analyzing and designing algorithms for several problems. Graph theory has a vital role in complex systems, especially in computer sciences. Applications of graph theory can be found in many scientific disciplines such as operational research, engineering, life sciences, management sciences, coding, and computer science. In the literature, there are two algorithms for constructing decompositions for the circulant graphs C2r,r with 2r vertices and r degree. For r,m⩾2, a circulant-balanced complete multipartite graph Cmr,(m-1)r is a generalization to C2r,r. Here, the major contributions are the generalization of the algorithms used for constructing the decompositions for the circulant graphs C2r,r to generalized algorithms that can be used for the decomposition of Cmr,(m-1)r that have mr vertices and (m-1)r degree. These generalized algorithms manage us to decompose the circulant graphs Cmr,(m-1)r by different graph classes. The paper's novelty is demonstrated by the fact that it is the first to suggest general approaches for decomposing circulant-balanced complete multipartite graphs. The constructed results in the present paper are suitable for generating different codes and also have great importance in the design of experiments.

Hamiltonian decompositions of 4-regular Cayley graphs of infinite abelian groups.

A well‐known conjecture of Alspach says that every 2k‐regular Cayley graph of a finite abelian group can be decomposed into Hamiltonian cycles. We consider an analogous question for infinite abelian groups. In this setting one natural analogue of a Hamiltonian cycle is a spanning double‐ray. However, a naive generalisation of Alspach's conjecture fails to hold in this setting due to the existence of 2k‐regular Cayley graphs with finite cuts F, where ∣F∣ and k differ in parity, which necessarily preclude the existence of a decomposition into spanning double‐rays. We show that every 4‐regular Cayley graph of an infinite abelian group all of whose finite cuts are even can be decomposed into spanning double‐rays, and so characterise when such decompositions exist. We also characterise when such graphs can be decomposed either into Hamiltonian circles, a more topological generalisation of a Hamiltonian cycle in infinite graphs, or into a Hamiltonian circle and a spanning double‐ray.

A Novel Approach for Cyclic Decompositions of Balanced Complete Bipartite Graphs into Infinite Graph Classes

Graph theory is considered an attractive field for finding the proof techniques in discrete mathematics. The results of graph theory have applications in many areas of social, computing, and natural sciences. Graph labelings and decompositions have received much attention in the literature. Several types of graph labeling were proposed for solving the problem of decomposing different graph classes. In the present paper, we propose a technique for labeling the vertices of a bipartite graph G with n edges, called orthogonal labeling, to yield cyclic decompositions of balanced complete bipartite graphs K n , n by the graph G . By applying the proposed orthogonal labeling technique, we had constructed decompositions of K n , n by paths, trees, one factorization, disjoint union of cycles, complete bipartite graphs, disjoint union of trees, caterpillars, and so forth. From the constructed results, we can confirm that the proposed orthogonal labeling technique is effective.

Countably determined ends and graphs

The directions of an infinite graph G are a tangle-like description of its ends: they are choice functions that choose a component of G−X for all finite vertex sets X⊆V(G) in a compatible manner.Although every direction is induced by a ray, there exist directions of graphs that are not uniquely determined by any countable subset of their choices. We characterise these directions and their countably determined counterparts in terms of star-like substructures or rays of the graph.Curiously, there exist graphs whose directions are all countably determined but which cannot be distinguished all at once by countably many choices.We structurally characterise the graphs whose directions can be distinguished all at once by countably many choices, and we structurally characterise the graphs whose directions cannot be distinguished in this manner. Our characterisations are phrased in terms of normal trees and tree-decompositions.Our four (sub)structural characterisations imply combinatorial characterisations of the four classes of infinite graphs that are defined by the first and second axiom of countability applied to their end spaces: the two classes of graphs whose end spaces are first countable or second countable, respectively, and the complements of these two classes.

Applications of Order Trees in Infinite Graphs

Traditionally, the trees studied in infinite graphs are trees of height at most ω, with each node adjacent to its parent and its children (and every branch of the tree inducing a path or a ray). However, there is also a method, systematically introduced by Brochet and Diestel, of turning arbitrary well-founded order trees T into graphs, in a way such that every T-branch induces a generalised path in the sense of Rado. This article contains an introduction to this method and then surveys four recent applications of order trees to infinite graphs, with relevance for well-quasi orderings, Hadwiger’s conjecture, normal spanning trees and end-structure, the last two addressing long-standing open problems by Halin.

Orthogonal Labeling for Some Different Infinite Graph Classes

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the mid 1960s. Graph labeling is one of the fascinating areas of graph theory with wide ranging applications. A collection $\mathcal{A}$ of isomorphic copies of a given subgraph $G$ of $T$ is said to be an orthogonal double cover (ODC) of the graph $T$ by $G$, if every edge of $T$ belongs to exactly two members of $\mathcal{A}$ and any two different elements from $\mathcal{A}$ share at most one edge. The existence of an orthogonal labeling of a graph $G$ with respect to a certain group implies the existence of the cyclic orthogonal double cover of the circulant graphs on that group. In this paper, we prove the existence of orthogonal labeling for some different infinite graph classes and hence, the existence of the cyclic orthogonal double cover of some different infinite circulant graphs.

The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph

Anantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient by itself to necessitate quantum ergodicity. We also show that having conditions of expansion and a specific relaxation of the high girth constraint present in later papers on quantum ergodicity is not sufficient. We do so by proving new properties of the Cartesian product of two graphs where one is infinite.

The Rigidity of Infinite Graphs II

The Rigidity of Infinite Graphs II

Bond topology of chain, ribbon and tube silicates. Part I. Graph-theory generation of infinite one-dimensional arrangements of (TO4)n- tetrahedra.

Chain, ribbon and tube silicates are based on one-dimensional polymerizations of (TO4)n- tetrahedra, where T = Si4+ plus P5+, V5+, As5+, Al3+, Fe3+ and B3+. Such polymerizations may be represented by infinite graphs (designated chain graphs) in which vertices represent tetrahedra and edges represent linkages between tetrahedra. The valence-sum rule of bond-valence theory limits the maximum degree of any vertex to 4 and the number of edges linking two vertices to 1 (corner-sharing tetrahedra). The unit cell (or repeat unit) of the chain graph generates the chain graph through action of translational symmetry operators. The (infinite) chain graph is converted into a finite graph by wrapping edges that exit the unit cell such that they link to vertices within the unit cell that are translationally equivalent to the vertices to which they link in the chain graph, and the wrapped graph preserves all topological information of the chain graph. A symbolic algebra is developed that represents the degree of each vertex in the wrapped graph. The wrapped graph is represented by its adjacency matrix which is modified to indicate the direction of wrapped edges, up (+c) or down (-c) along the direction of polymerization. The symbolic algebra is used to generate all possible vertex connectivities for graphs with ≤8 vertices. This method of representing chain graphs by finite matrices may now be inverted to generate all non-isomorphic chain graphs with ≤8 vertices for all possible vertex connectivities. MatLabR2019b code is provided for computationally intensive steps of this method and ∼3000 finite graphs (and associated adjacency matrices) and ∼1500 chain graphs are generated.