We prove that an ultrametric space can be bi-Lipschitz embedded in R if its metric dimension in Assouad’s sense is smaller than n. We also characterize ultrametric spaces up to bi-Lipschitz homeomorphism as dense subspaces of ultrametric inverse limits of certain inverse sequences of discrete spaces.

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, otherwise F has unbou...

The metric dimension of a graph G is the minimum number of vertices in a subset S of the vertex set of G such that all other vertices are uniquely determined by their distances to the vertices in S. In this paper we investigate the metric dimension of the random graph G(n, p) for a wide range of probabilities p = p(n).

A technique of spatial-spectral quantization of hyperspectral images is introduced. Thus a quantized hyperspectral image is just summarized by K spectra which represent the spatial and spectral structures of the image. The proposed technique is based on α−connected components on a region adjacency graph. The main ingredient is a dissimilarity metric. In order to choose the metric that best fit ...

Trustworthiness is a field of research that seeks to estimate the credibility of information by using knowledge of the source of the information. The most interesting form of this problem is when different pieces of information share sources, and when there is conflicting information from different sources. This model can be naturally represented as a bipartite graph. In order to understand thi...

SOME RESULTS ON DISCREPANCIES BETWEEN METRIC DIMENSION AND PARTITION DIMENSION OF A GRAPH* Muhammad Imran Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan [email protected] ABSTRACT. In this paper some infinite regular graphs generated by tilings of the plane by infinite hexagonal grid are considered. It is prove...

In the previous lecture notes, we saw that any metric (X, d) with |X| = n can be embedded into R 2 n) under any the `1 metric (actually, the same embedding works for any `p metic), with distortion O(log n). Here, we describe an extremely useful approach for reducing the dimensionality of a Euclidean (`2) metric, while incurring very little distortion. Such dimension reduction is useful for a nu...