Graph substitutions

1998 ◽  
Vol 18 (3) ◽  
pp. 661-685 ◽  
Author(s):  
JOSEPH P. PREVITE
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Infinite Graphs ◽  
Complex Objects ◽  
Finite Graphs ◽  
Multicellular Organisms ◽  
Basic Graph ◽  
Replacement Rule

In 1984, Gromov (see [4] and [6]) introduced the idea of subdividing a ‘branching’ polyhedron into smaller cells and replacing these cells by more complex objects, reminiscent of the growth of multicellular organisms in biology. The simplest situation of this kind is a graph substitution which replaces certain subgraphs in a graph $G$ by bigger finite graphs. The most basic graph substitution is a vertex replacement rule which replaces certain vertices of $G$ with finite graphs. This paper develops a framework for studying vertex replacements and discusses the asymptotic behavior of iterated vertex replacements, the limit objects, and the induced dynamics on the space of infinite graphs from the viewpoint of geometry and dynamical systems.

scholarly journals Liftings in Finite Graphs and Linkages in Infinite Graphs with Prescribed Edge-Connectivity

2016 ◽  
Vol 32 (6) ◽  
pp. 2575-2589
Author(s):  
Seongmin Ok ◽  
R. Bruce Richter ◽  
Carsten Thomassen
Keyword(s):  
Infinite Graphs ◽  
Edge Connectivity ◽  
Finite Graphs

scholarly journals On Ramsey-Minimal Infinite Graphs

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Jordan Barrett ◽  
Valentino Vito
Keyword(s):  
Ramsey Theory ◽  
Infinite Graphs ◽  
Minimal Graph ◽  
Finite Graphs ◽  
Minimal Graphs ◽  
Finite Case ◽  
Ramsey Minimal Graph ◽  
General Conditions

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$. We generalize this study to infinite graphs $G$, $H$; in particular, we want to determine if there is a minimal such $F$. This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves. We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph. We use these to prove, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \geqslant 1$ is a matching, then the pair $(S_\infty,nK_2)$ admits no Ramsey-minimal graphs.

scholarly journals Topological Infinite Gammoids, and a New Menger-Type Theorem for Infinite Graphs

10.37236/6083 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Johannes Carmesin
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Infinite Graphs ◽  
Disjoint Paths ◽  
Natural Topology ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Vertex Sets ◽  
Special Case

Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid.As our main tool, we prove for any infinite graph $G$ with vertex-sets $A$ and $B$, if every finite subset of $A$ is linked to $B$ by disjoint paths, then the whole of $A$ can be linked to the closure of $B$ by disjoint paths or rays in a natural topology on $G$ and its ends.This latter theorem implies the topological Menger theorem of Diestel for locally finite graphs. It also implies a special case of the infinite Menger theorem of Aharoni and Berger.

scholarly journals Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Thierry Horsin ◽  
Mohamed Ali Jendoubi
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Equilibrium Points ◽  
Infinite Dimensional ◽  
Infinite Dimensional Dynamical Systems ◽  
Finite Dimensional ◽  
Angle Condition ◽  
Non Local ◽  
Second Order Systems

<p style='text-indent:20px;'>In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.</p>

RAPID GROWTH PATHS IN CONVEX-VALUED RANDOM DYNAMICAL SYSTEMS

2001 ◽  
Vol 01 (04) ◽  
pp. 493-509 ◽  
Author(s):  
IGOR V. EVSTIGNEEV ◽  
MICHAEL I. TAKSAR
Keyword(s):  
Economic Growth ◽  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Growth Rates ◽  
Stochastic Models ◽  
Rapid Growth ◽  
Random Dynamical Systems ◽  
Random Vectors ◽  
Time Period

This paper examines set-valued random dynamical systems defined by convex homogeneous stochastic operators. The operators under consideration transform elements of a cone contained in a space of random vectors into subsets of the cone. We study rapid paths of such dynamical systems, i.e. those paths which maximize (appropriately defined) growth rates at every time period. Questions of existence, uniqueness and asymptotic behavior of infinite rapid trajectories are considered. The study is motivated by problems related to stochastic models of economic growth.

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