A metric basis is a set W of vertices of a graph G(V,E) such that for every pair of vertices u, v of G, there exists a vertex w ∈ W with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. The minimum cardinality of a metric basis for G is called the metric dimension. A pair of vertices u, v is said to be strongly resolved by...

Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs Gn : F = (Gn)n≥1 depending on n as follows: the order |V (G)| = φ(n) and lim n→∞ φ(n) = ∞ . If there exists a constant C > 0 such that dim(Gn) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension. If all graphs in F hav...

Resolving parameters are a fundamental area of combinatorics with applications not only to many branches but also other sciences. In this study, we construct class Toeplitz graphs and will be denoted by T 2 n W ...

In this paper we extend the definitions of mean dimension and metric di-mension for non-autonomous dynamical systems. We show some properties extension furthermore applications to single continuous maps.

Recall that the asymptotic dimension is a coarse geometric analogue of the covering dimension in topology [14]. More precisely, the asymptotic dimension for a metric space is the smallest integer n such that for any r > 0, there exists a uniformly bounded cover C = {Ui}i∈I of the metric space for which the rmultiplicity of C is at most n + 1, i.e. no ball of radius r in the metric space interse...

Given a set of classifiers and a probability distribution over their domain, one can define a metric by taking the distance between a pair of classifiers to be the probability that they classify a random item differently. We prove bounds on the sample complexity of PAC learning in terms of the doubling dimension of this metric. These bounds imply known bounds on the sample complexity of learnin...