Compactness and finite forcibility of graphons

Graphons are analytic objects associated with convergent sequences of dense graphs. Problems from extremal combinatorics and theoretical computer science led to a study of finitely forcible graphons, i.e., graphons determined by finitely many subgraph densities. Lovasz and Szegedy conjectured that the topological space of typical vertices of such a graphon always has finite dimension, which would also have implications on the number of parts in its weak regular partition. We disprove the conjecture by constructing a finitely forcible graphon with the space of typical vertices with infinite dimension.

Journal of Combinatorial Theory, Series B, 2015

We investigate when limits of graphs (graphons) and permutations (permutons) are uniquely determined by finitely many densities of their substructures, i.e., when they are finitely forcible. Every permuton can be associated with a graphon through the notion of permutation graphs. We find permutons that are finitely forcible but the associated graphons are not. We also show that all permutons that can be expressed as a finite combination of monotone permutons and quasirandom permutons are finitely forcible, which is the permuton counterpart of the result of Lovász and Sós for graphons.

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.

Opuscula Mathematica

A simple graph G is called a compact graph if G contains no isolated vertices and for each pair x, y of non-adjacent vertices of G, there is a vertex z with N (x) ∪ N (y) ⊆ N (z), where N (v) is the neighborhood of v, for every vertex v of G. In this paper, compact graphs with sufficient number of edges are studied. Also, it is proved that every regular compact graph is strongly regular. Some results about cycles in compact graphs are proved, too. Among other results, it is proved that if the ascending chain condition holds for the set of neighbors of a compact graph G, then the descending chain condition holds for the set of neighbors of G.

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.

Transactions of the American Mathematical Society, 1993

We consider a connected graph, having countably infinite vertex set X X , which is permitted to have vertices of infinite degree. For a transient irreducible transition matrix P P corresponding to a nearest neighbor random walk on X X , we study the associated harmonic functions on X X and, in particular, the Martin compactification. We also study the end compactification of the graph. When the graph is a tree, we show that these compactifications coincide; they are a disjoint union of X X , the set of ends, and the set of improper vertices—new points associated with vertices of infinite degree. Other results proved include a solution of the Dirichlet problem in the context of the end compactification of a general graph. Applications are given to, e.g., the Cayley graph of a free group on infinitely many generators.

2014

In the paper, we consider Hilbert spaces of functions on infinite graphs, and their compactifications. We arrive at a sampling formula in the spirit of Shannon; the idea is that we allow for sampling of functions f defined on a continuum completion of an infinite graph G, sampling the continuum by values of f at points in the graph G. Rather than the more traditional frequency analysis of band-limited functions from Shannon, our analysis is instead based on reproducing kernel Hilbert spaces built from a prescribed infinite system of resistors on G.

Fundamenta Mathematicae, 2003

For any class K of compacta and any compactum X we say that: (a) X is confluently K-representable if X is homeomorphic to the inverse limit of an inverse sequence of members of K with confluent bonding mappings, and (b) X is confluently K-like provided that X admits, for every ε > 0, a confluent ε-mapping onto a member of K. The symbol LC stands for the class of all locally connected compacta. It is proved in this paper that for each compactum X and each family K of graphs, X is confluently Krepresentable if and only if X is confluently K-like. We also show that for any compactum the properties of: (1) being confluently graph-representable, and (2) being 1-dimensional and confluently LC-like, are equivalent. Consequently, all locally connected curves are confluently graph-representable. We also conclude that all confluently arc-like continua are homeomorphic to inverse limits of arcs with open bonding mappings, and all confluently tree-like continua are absolute retracts for hereditarily unicoherent continua.

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.

Topology and its Applications, 1998

We have collected several open problems on graphs which arise in geometric topology, in particular in the following areas: (1) basic embeddability of compacta into the plane R'; (2) approximability of maps by embeddings: (3) uncountable collections of continua in IL?* and their span; and (4) representations of closed PL manifolds by colored graphs. These problems should be of interest to both topologists and combinatorists. 0 1998 Elsevier Science B.V.