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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| Language: | English |
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MDPI AG
2025-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/3/462 |
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| author | Yiming Lei Zhongrui Wang Bing Dai |
| author_facet | Yiming Lei Zhongrui Wang Bing Dai |
| author_sort | Yiming Lei |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-d764733030ba4e3fbe60649ce77c0773 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-d764733030ba4e3fbe60649ce77c07732025-08-20T02:48:06ZengMDPI AGMathematics2227-73902025-01-0113346210.3390/math13030462Metric Dimensions of Metric Spaces over Vector GroupsYiming Lei0Zhongrui Wang1Bing Dai2College of Mathematical Sciences, Bohai University, Jinzhou 121013, ChinaCollege of Mathematical Sciences, Bohai University, Jinzhou 121013, ChinaCollege of Mathematical Sciences, Bohai University, Jinzhou 121013, ChinaLet <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.https://www.mdpi.com/2227-7390/13/3/462Cayley graphmetric dimensionmetric spacevector group |
| spellingShingle | Yiming Lei Zhongrui Wang Bing Dai Metric Dimensions of Metric Spaces over Vector Groups Mathematics Cayley graph metric dimension metric space vector group |
| title | Metric Dimensions of Metric Spaces over Vector Groups |
| title_full | Metric Dimensions of Metric Spaces over Vector Groups |
| title_fullStr | Metric Dimensions of Metric Spaces over Vector Groups |
| title_full_unstemmed | Metric Dimensions of Metric Spaces over Vector Groups |
| title_short | Metric Dimensions of Metric Spaces over Vector Groups |
| title_sort | metric dimensions of metric spaces over vector groups |
| topic | Cayley graph metric dimension metric space vector group |
| url | https://www.mdpi.com/2227-7390/13/3/462 |
| work_keys_str_mv | AT yiminglei metricdimensionsofmetricspacesovervectorgroups AT zhongruiwang metricdimensionsofmetricspacesovervectorgroups AT bingdai metricdimensionsofmetricspacesovervectorgroups |