Graph Theoretical Models of Discrete Spaces with Locally Non-spherical Topology

Alexander V. Evako^1,

¹“Dianet”, Laboratory of Digital Technologies, Moscow, Russia

Cite this paper:
Alexander V. Evako. Graph Theoretical Models of Discrete Spaces with Locally Non-spherical Topology. Applied Mathematics and Physics. 2017; 5(2):47-52. doi: 10.12691/amp-5-2-3

Abstract

A graph theoretical model of a continuous space is a graph with the same topological structure as its continuous counterpart. A digital closed n-dimensional manifold with a locally spherical topology is a graph theoretic model for a continuous closed n-dimensional manifold. This paper defines and studies properties of a new class of digital n-dimensional spaces with a locally non-spherical topology. We prove that such spaces have the dimension n≥3. We define and investigate properties of digital 3- and 5-dimensional closed surfaces with a local toroidal and projective plane topology. These spaces have no direct continuous counterparts among n-dimensional manifolds in classical topology. These results arise questions like what physical, chemical or biological structures can be described by digital n-dimensional surfaces with a locally non-spherical topology.

Keywords:
graph manifold discrete space non-spherical topology torus projective plane

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