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#### Article Title

#### Abstract

Let $(X, d)$ be an unbounded metric space and let $\tilde r=(r_n)_{n\in\mathbb N}$ be a sequence of positive real numbers tending to infinity. A pretangent space $\Omega_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity is a limit of the rescaling sequence $\left(X, \frac{1}{r_n}d\right).$ The set of all pretangent spaces $\Omega_{\infty, \tilde r}^{X}$ is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph $(G_{X, \tilde r}, \rho_{X})$ whose maximal cliques coincide with $\Omega_{\infty, \tilde r}^{X}$ and the weight $\rho_{X}$ is defined by metrics on $\Omega_{\infty, \tilde r}^{X}$. We describe the structure of metric spaces having finite asymptotic clusters of pretangent spaces and characterize the finite weighted graphs which are isomorphic to these clusters.

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This work is licensed under a Creative Commons Attribution 4.0 License.

#### Recommended Citation

Bilet, Viktoriia and Dovgoshey, Oleksiy
(2018)
"Finite Asymptotic Clusters of Metric Spaces,"
*Theory and Applications of Graphs*: Vol. 5
:
Iss.
2
, Article 1.

DOI: 10.20429/tag.2018.050201

Available at:
https://digitalcommons.georgiasouthern.edu/tag/vol5/iss2/1