Infinite Graphs Research Articles

Bootstrap Percolation in Strong Products of Graphs

Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G,r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider percolation numbers of strong products of graphs. If $G$ is the strong product $G_1\boxtimes \cdots \boxtimes G_k$ of $k$ connected graphs, we prove that $m(G,r)=r$ as soon as $r\le 2^{k-1}$ and $|V(G)|\ge r$. As a dichotomy, we present a family of strong products of $k$ connected graphs with the $(2^{k-1}+1)$-percolation number arbitrarily large. We refine these results for strong products of graphs in which at least two factors have at least three vertices. In addition, when all factors $G_i$ have at least three vertices we prove that $m(G_1 \boxtimes \dots \boxtimes G_k,r)\leq 3^{k-1} -k$ for all $r\leq 2^k-1$, and we again get a dichotomy, since there exist families of strong products of $k$ graphs such that their $2^{k}$-percolation numbers are arbitrarily large. While $m(G\boxtimes H,3)=3$ if both $G$ and $H$ have at least three vertices, we also characterize the strong prisms $G\boxtimes K_2$ for which this equality holds. Some of the results naturally extend to infinite graphs, and we briefly consider percolation numbers of strong products of two-way infinite paths.

BOREL LINE GRAPHS

Abstract We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the nine finite graphs from the classical result of Beineke together with a 10th infinite graph associated with the equivalence relation $\mathbb {E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman–Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

Capacity of infinite graphs over non-Archimedean ordered fields

In this article we study the notion of capacity of a vertex for infinite graphs over non-Archimedean fields. In contrast to graphs over the real field monotone limits do not need to exist. Thus, in our situation next to positive and null capacity there is a third case of divergent capacity. However, we show that either of these cases is independent of the choice of the vertex and is therefore a global property for connected graphs. The capacity is shown to connect the minimization of the energy, solutions of the Dirichlet problem and existence of a Green's function. We furthermore give sufficient criteria in form of a Nash-Williams test, study the relation to Hardy inequalities and discuss the existence of positive superharmonic functions. Finally, we investigate the analytic features of the transition operator in relation to the inverse of the Laplace operator.

Towards Nash‐Williams orientation conjecture for infinite graphs

AbstractIn 1960 Nash‐Williams proved that an edge‐connectivity of is sufficient for a finite graph to have a ‐arc‐connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on the auxiliary lifting graph, proved that ‐edge‐connected infinite graphs admit a ‐arc‐connected orientation. Here we improve this result for the class of one‐ended locally finite graphs and show that an edge‐connectivity of is enough in that case. Crucial to this improvement are results presented in a separate paper, by the same author of this paper, on the key concept of the lifting graph, extending results by Ok, Richter, and Thomassen.

Metric basis and metric dimension of some infinite planar graphs

Let [Formula: see text] be a nontrivial connected simple graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The metric dimension of a graph [Formula: see text], denoted by dim([Formula: see text]), refers to the smallest set of vertices required to uniquely identify every vertex in the graph [Formula: see text]. A family of simple connected graphs say [Formula: see text], where [Formula: see text] has a constant metric dimension, if dim[Formula: see text] is finite and does not depend on the choice of [Formula: see text] in [Formula: see text]. In this paper, we consider two infinite families of planar graphs, say [Formula: see text], where [Formula: see text] and [Formula: see text], where [Formula: see text], and investigate their metric basis as well as the metric dimension. Additionally, we prove that the metric basis for these two graphs are independent.

Note on intrinsic metrics on graphs

AbstractWe study the set of intrinsic metrics on a given graph. This is a convex compact set and it carries a natural order. We investigate existence of largest elements with respect to this order. We show that the only locally finite graphs which admit a largest intrinsic metric are certain finite star graphs. In particular, all infinite locally finite graphs do not admit a largest intrinsic metric. For infinite graphs which are not locally finite the set of intrinsic metrics may be trivial as we show by an example. Moreover, we give a characterization for the existence of intrinsic metrics with finite balls for weakly spherically symmetric graphs.

Continuously indexed graphical models

Abstract Let X={Xu}u∈U be a real-valued Gaussian process indexed by a set U. We show that X can be viewed as a graphical model with an uncountably infinite graph, where each Xu is a vertex. This graph is characterized by the reproducing property of X’s covariance kernel, without restricting U to be finite or countable, allowing the modelling of stochastic processes in continuous time/space. Unlike traditional methods, this characterization is not based on zero entries of an inverse covariance, posing challenges for structure estimation. We propose a plug-in methodology that targets graph recovery up to a finite resolution and shows consistency for graphs which are sufficiently regular and that can be applied to virtually any measurement regime. Furthermore, we derive convergence rates and finite-sample guarantees for the method, and demonstrate its performance through a simulation study and two data analyses.

Minimum Spanning Trees in Infinite Graphs: Theory and Algorithms

Minimum Spanning Trees in Infinite Graphs: Theory and Algorithms

On vertex‐transitive graphs with a unique hamiltonian cycle

AbstractA graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex‐transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group has a Cayley graph of degree that has a unique hamiltonian circle. (A weaker statement had been conjectured by Georgakopoulos.) Furthermore, we prove that these Cayley graphs of are outerplanar.

Parabolic Schrödinger networks

A Schrödinger network Nq={X,ζ,q} is an infinite graph with a set of vertices X and edges that are countably infinite, a transition function ζ(a,b)≥0 and a function q(a) on X. A perturbed Laplace operator Δq is defined as Δql(a)=Δl(a)−q(a)l(a). A solution h(a) to the equation Δqh(a)=0 is termed a q-average function, and a q-superaverage function v(a) satisfies Δqv(a)≤0. The network is called q-hyperbolic if there exists a function s(a)>0, q-superaverage on X; otherwise, it is a q-parabolic network, also mentioned as a parabolic Schrödinger network. In this paper, we develop a theory of function on parabolic Schrödinger networks by using potential-theoretic techniques associated with q-superaverage and q-average functions, which are similar to the function theory on parabolic Riemannian manifolds.

The asynchronous DeGroot dynamics

AbstractWe analyze the asynchronous version of the DeGroot dynamics: In a connected graph with nodes, each node has an initial opinions in and an independent Poisson clock. When a clock at a node rings, the opinion at is replaced by the average opinion of its neighbors. It is well known that the opinions converge to a consensus. We show that the expected time to reach ‐consensus is poly in undirected graphs and in Eulerian digraphs, but for some digraphs of bounded degree it is exponential. Our main result is that in undirected graphs and Eulerian digraphs, if the degrees are uniformly bounded and the initial opinions are i.i.d., then for every fixed . We give sharp estimates for the variance of the limiting consensus opinion, which measures the ability to aggregate information (“wisdom of the crowd”). We also prove generalizations to non‐reversible Markov chains and infinite graphs. New results of independent interest on fragmentation processes and coupled random walks are crucial to our analysis.

A Liouville theorem for elliptic equations with a potential on infinite graphs

We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is u≡0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u\\equiv 0$$\\end{document}. We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions.

On the optimality and decay of p-Hardy weights on graphs

We construct optimal Hardy weights to subcritical energy functionals h associated with quasilinear Schrödinger operators on infinite graphs. Here, optimality means that the weight w is the largest possible with respect to a partial ordering, and that the corresponding shifted energy functional h-w\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h-w$$\\end{document} is null-critical. Moreover, we show a decay condition of Hardy weights in terms of their integrability with respect to certain integral weights. As an application of the decay condition, we show that null-criticality implies optimality near infinity. We also briefly discuss an uncertainty-type principle, a Rellich-type inequality and examples.

Some spectral comparison results on infinite quantum graphs

In this paper we establish spectral comparison results for Schrödinger operators on a certain class of infinite quantum graphs, using recent results obtained in the finite setting. We also show that new features do appear on infinite quantum graphs such as a modified local Weyl law. In this sense, we regard this paper as a starting point for a more thorough investigation of spectral comparison results on more general infinite metric graphs.

Maximizing the Average Environmental Benefit of a Fleet of Drones under a Periodic Schedule of Tasks

Unmanned aerial vehicles (UAVs, drones) are not just a technological achievement based on modern ideas of artificial intelligence; they also provide a sustainable solution for green technologies in logistics, transport, and material handling. In particular, using battery-powered UAVs to transport products can significantly decrease energy and fuel expenses, reduce environmental pollution, and improve the efficiency of clean technologies through improved energy-saving efficiency. We consider the problem of maximizing the average environmental benefit of a fleet of drones given a periodic schedule of tasks performed by the fleet of vehicles. To solve the problem efficiently, we formulate it as an optimization problem on an infinite periodic graph and reduce it to a special type of parametric assignment problem. We exactly solve the problem under consideration in O(n3) time, where n is the number of flights performed by UAVs.

Spectral properties of Sturm–Liouville operators on infinite metric graphs

Spectral properties of Sturm–Liouville operators on infinite metric graphs

Coarse Geometry of Quasi-Transitive Graphs Beyond Planarity

We study geometric and topological properties of infinite graphs that are quasi-isometric to a planar graph of bounded degree. We prove that every locally finite quasi-transitive graph excluding a minor is quasi-isometric to a planar graph of bounded degree. We use the result to give a simple proof of the result that finitely generated minor-excluded groups have Assouad-Nagata dimension at most 2 (this is known to hold in greater generality, but all known proofs use significantly deeper tools). We also prove that every locally finite quasi-transitive graph that is quasi-isometric to a planar graph is $k$-planar for some $k$ (i.e. it has a planar drawing with at most $k$ crossings per edge), and discuss a possible approach to prove the converse statement.

Non-triviality of the phase transition for percolation on finite transitive graphs

We prove that if (G_{n})_{n \geq1}=((V_{n},E_{n}))_{n\geq 1} is a sequence of finite, vertex-transitive graphs with bounded degrees and |V_{n}|\to\infty that is at least (1+\varepsilon) -dimensional for some \varepsilon>0 in the sense that \operatorname{diam} (G_{n})=O(|V_{n}|^{1/(1+\varepsilon)}) \quad \text{as }n\to\infty then this sequence of graphs has a non-trivial phase transition for Bernoulli bond percolation. More precisely, we prove under these conditions that for each 0<\alpha <1 there exists p_{c}(\alpha)<1 such that for each p\geq p_{c}(\alpha) , Bernoulli- p bond percolation on G_{n} has a cluster of size at least \alpha |V_{n}| with probability tending to 1 as n\to \infty . In fact, we prove more generally that there exists a universal constant a such that the same conclusion holds whenever \operatorname{diam} (G_{n})=O\Big(\frac{|V_{n}|}{(\log |V_{n}|)^{a}}\Big) \quad \text{as }n\to\infty. This verifies a conjecture of Benjamini (2001) up to the value of the constant a , which he suggested should be 1 . We also prove a generalization of this result to quasitransitive graph sequences with a bounded number of vertex orbits. A key step in our argument is a direct proof of our result when the graphs G_{n} are all Cayley graphs of Abelian groups, in which case we show that one may indeed take a=1 . This result relies crucially on a new theorem of independent interest stating roughly that balls in arbitrary Abelian Cayley graphs can always be approximated by boxes in \Z^{d} with the standard generating set. Another key step is to adapt the methods of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin [Duke Math. J. 169 (2020)] from infinite graphs to finite graphs. This adaptation also leads to an isoperimetric criterion for infinite graphs to have a non-trivial uniqueness phase (i.e., to have p_{u}<1 ), which is of independent interest. We also prove that the set of possible values of the critical probability of an infinite quasitransitive graph has a gap at 1 in the sense that for every k,n<\infty there exists \varepsilon>0 such that every infinite graph G of degree at most k whose vertex set has at most n orbits under \operatorname{Aut}(G) has either p_c=1 or p_c\leq 1-\varepsilon .

Bond topology of chain, ribbon and tube silicates. Part II. Geometrical analysis of infinite 1D arrangements of (TO4)n- tetrahedra.

In Part I of this series, all topologically possible 1-periodic infinite graphs (chain graphs) representing chains of tetrahedra with up to 6-8 vertices (tetrahedra) per repeat unit were generated. This paper examines possible restraints on embedding these chain graphs into Euclidean space such that they are compatible with the metrics of chains of tetrahedra in observed crystal structures. Chain-silicate minerals with T = Si4+ (plus P5+, V5+, As5+, Al3+, Fe3+, B3+, Be2+, Zn2+ and Mg2+) have a grand nearest-neighbour ⟨T-T⟩ distance of 3.06±0.15 Å and a minimum T...T separation of 3.71 Å between non-nearest-neighbour tetrahedra, and in order for embedded chain graphs (called unit-distance graphs) to be possible atomic arrangements in crystals, they must conform to these metrics, a process termed equalization. It is shown that equalization of all acyclic chain graphs is possible in 2D and 3D, and that equalization of most cyclic chain graphs is possible in 3D but not necessarily in 2D. All unique ways in which non-isomorphic vertices may be moved are designated modes of geometric modification. If a mode (m) is applied to an equalized unit-distance graph such that a new geometrically distinct unit-distance graph is produced without changing the lengths of any edges, the mode is designated as valid (mv); if a new geometrically distinct unit-distance graph cannot be produced, the mode is invalid (mi). The parameters mv and mi are used to define ranges of rigidity of the unit-distance graphs, and are related to the edge-to-vertex ratio, e/n, of the parent chain graph. The program GraphT-T was developed to embed any chain graph into Euclidean space subject to the metric restraints on T-T and T...T. Embedding a selection of chain graphs with differing e/n ratios shows that the principal reason why many topologically possible chains cannot occur in crystal structures is due to violation of the requirement that T...T > 3.71 Å. Such a restraint becomes increasingly restrictive as e/n increases and indicates why chains with stoichiometry TO<2.5 do not occur in crystal structures.

On the Korteweg‐de Vries approximation for a Boussinesq equation posed on the infinite necklace graph

Motivated by the question how to describe long‐wave dynamics on periodic networks, we consider a Boussinesq equation posed on the infinite periodic necklace graph. For the description of long‐wave traveling waves, we derive the KdV equation and establish the validity of this formal approximation by providing estimates for the error. The proof is based on suitable energy estimates. As a consequence of the approximation result, the soliton dynamics present in the KdV equation can approximately be seen for the original system, too. There are no serious obstacles to transfer the analysis to other dispersive systems or to other periodic quantum graphs for which the KdV equation can be derived. However, a general KdV validity theory on quantum graphs would be extremely abstract and of minor use.