Metric Dimensions of Metric Spaces over Vector Groups

Metric Dimensions of Metric Spaces over Vector Groups

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>ρ</mi><mo>)</mo></mrow></semantics></math>...

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Bibliographic Details
Main Authors:	Yiming Lei, Zhongrui Wang, Bing Dai
Format:	Article
Language:	English
Published:	MDPI AG 2025-01-01
Series:	Mathematics
Subjects:	Cayley graph metric dimension metric space vector group
Online Access:	https://www.mdpi.com/2227-7390/13/3/462
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Summary:	Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>ρ</mi><mo>)</mo></mrow></semantics></math></inline-formula> be a metric space. A subset <i>A</i> of <i>X</i> resolves <i>X</i> if every point <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula> is uniquely identified by the distances <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ρ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></semantics></math></inline-formula>. The metric dimension of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>ρ</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the minimum integer <i>k</i> for which a set <i>A</i> of cardinality <i>k</i> resolves <i>X</i>. We consider the metric spaces of Cayley graphs of vector groups over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">Z</mi></semantics></math></inline-formula>. It was shown that for any generating set <i>S</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">Z</mi></semantics></math></inline-formula>, the metric dimension of the metric space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><mi>X</mi><mo>(</mo><mi mathvariant="double-struck">Z</mi><mo>,</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> is, at most, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>max</mi><mi>S</mi></mrow></semantics></math></inline-formula>. Thus, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><mi>X</mi><mo>(</mo><mi mathvariant="double-struck">Z</mi><mo>,</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> can be resolved by a finite set. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. We show that for any finite generating set <i>S</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">Z</mi><mi>n</mi></msup></semantics></math></inline-formula>, the metric space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><mi>X</mi><mo>(</mo><msup><mi mathvariant="double-struck">Z</mi><mi>n</mi></msup><mo>,</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> cannot be resolved by a finite set.
ISSN:	2227-7390