Inverse Limits and Topologies of Infinite Graphs

Mathematical Proceedings of the Cambridge Philosophical Society, 2001

There are different definitions of ends in non-locally-finite graphs which are all equivalent in the locally finite case. We prove the compactness of the end-topology that is based on the principle of removing finite sets of vertices and give a proof of the compactness of the end-topology that is constructed by the principle of removing finite sets of edges. For the latter case there exists already a proof in [1], which only works on graphs with countably infinite vertex sets and in contrast to which we do not use the Theorem of Tychonoff. We also construct a new topology of ends that arises from the principle of removing sets of vertices with finite diameter and give applications that underline the advantages of this new definition.

International Journal of Mathematical Analysis

The aim of this article is to introduce a new approach of applying the topology on digraphs by associate two topologies with the set of edges of any directed graph, called compatible and incompatible edges topologies. Some properties of these topologies were investigated. In particular, the properties of connectivity and density are studied. Giving fundamental step toward studying some properties of directed graphs by their corresponding topology, which is introduced in this article, is our motivation.

Mathematics and Statistics, 2024

The primary aim of this paper is to establish and analyze certain topological structures linked with a specified graph G. In a graph G, a vertex u is considered a neighbor of another vertex v if there exists an edge uv in G. Furthermore, we define two vertices (or edges) in G as coneighbors if they share identical sets of neighboring vertices (or edges). The topology under consideration arises from the collections of vertices that are coneighbor and the collections of edges that are coneighbor within the graph. It is proved that the coneighbor topology of every non-coneighbor graph is homeomorphic to the included point topology while this space is quasi-discrete if and only if the graph contains at least one coneighbor set of vertices and some examples of coneighbor topologies of special graphs are presented to be quasi-discrete spaces such as (a path, a cycle and a bipartite) graphs. Moreover, several topological properties of the coneighbor space are presented. We proved that the coneighbor topological space associated with a graph G always has dimension one and satisfies the T 1/2 axiom. Also, the family of θ-open sets is determined in this spaces and it is proved that this space is almost compact whenever the family of coneighbor sets is finite. Finally, we looked at some graphs in which the coneighbor space fulfills other topological concepts such as connectedness, compactness and countable compactness.

Bulletin of The Iranian Mathematical Society, 2013

Let G = (V;E) be a locally nite graph, i.e. a graph in which every vertex has nitely many adjacent vertices. In this paper, we associate a topology to G, called graphic topology of G and we show that it is an Alexandro topology, i.e. a topology in which intersection of each family of open sets is open. Then we investigate some properties of this topology. Our motivation is to give an elementary step toward investigation of some properties of locally nite graphs by their corresponding topology which we introduce in this paper.

Topology and its Applications, 1990

@n.~ graph is defined to be a compact, connected, locally connected metric space which is not separated into more than n components by any subcontinuum and no subcontinuum is separated into more than L components by any of its subcontinua. If X is a 8,,., graph andj is a continuous surjection of X onto X, then the inverse limit space (X,/) is a O,, continuum (not necessarily locally connected). Furthermore (X,f) admits a unique minimal monotone, upper semicontinuous decomposition 3 such that the quotient space (X,j)/9 is a O,,., graph if and only if (X,/) contains no indecomposable subcontinua with nonempty interior.

Discrete Mathematics, 1992

PrCa, P., Graphs and topologies on discrete sets, Discrete Mathematics 103 (1992) 189-197. We show that a graph admits a topology on its node set which is compatible with the usual connectivity of undirected graphs if, and only if, it is a comparability graph. Then, we give a similar condition for the weak connectivity of oriented graphs and show there is no topology which is compatible with the strong connectivity of oriented graphs. We also give a necessary and sufficient condition for a topology on a discrete set to be 'representable' by an undirected graph. R&sum6 Nous montrons qu'un graphe admet une topologie sur I'ensemble de ses sommets compatible avec la connexit6 usuelle des graphes non-orient& si, et settlement si c'est un graphe de comparabilitt; puis nous donnons une condition similaire pour la connexite faible des graphes orient& et montrons la non-existence d'une topologie compatible avec la connexite forte. Nous donnons Cgalement une condition necessaire et suffisante pour qu'une topologie sur un ensemble discret soit 'representable' par un graphe non-oriente.

PLOS ONE

In this paper, we study the limit of compactness which is a graph index originally introduced for measuring structural characteristics of hypermedia. Applying compactness to large scale small-world graphs (Mehler, 2008) observed its limit behaviour to be equal 1. The striking question concerning this finding was whether this limit behaviour resulted from the specifics of small-world graphs or was simply an artefact. In this paper, we determine the necessary and sufficient conditions for any sequence of connected graphs resulting in a limit value of C B = 1 which can be generalized with some consideration for the case of disconnected graph classes (Theorem 3). This result can be applied to many well-known classes of connected graphs. Here, we illustrate it by considering four examples. In fact, our proof-theoretical approach allows for quickly obtaining the limit value of compactness for many graph classes sparing computational costs.

Fundamenta Mathematicae, 2005

We provide a method of construction of topologically mixing maps f on topological graph G with the shadowing property and such that the inverse limit with f as the single bonding map is a hereditarily indecomposable continuum. Additionally, f can be obtained as an arbitrarily small perturbation of any given topologically exact map on G, and if G is the unit circle, then f is necessarily topologically exact.

2013

The graph topology τ Γ is the topology on the space C(X) of all continuous functions defined on a Tychonoff space X inherited from the Vietoris topology on X × R after identifying continuous functions with their graphs. It is shown that all completeness properties between complete metrizability and hereditary Baireness coincide for the graph topology if and only if X is countably compact; however, the graph topology is α-favorable in the strong Choquet game, regardless of X. Analogous results are obtained for the fine topology on C(X). Pseudocompleteness, along with properties related to 1st and 2nd countability of (C(X), τ Γ) are also investigated.

AIMS Mathematics, 2020

In this paper, we study topologies generated by simple undirected graphs without isolated vertices and their properties. We generate firstly a topology using a simple undirected graph without isolated vertices. Moreover, we investigate properties of the topologies generated by certain graphs. Finally,we present continuity and opennes of functions defined from one graph to another via the topologies generated by the graphs. From this point of view, we present necessary and sufficient condition for the topological spaces generated by two different graphs to be homeomorphic.