The Structural Properties of (2, 6)-Fullerenes
A <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math>&...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-11-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/15/11/2078 |
| _version_ | 1797457627455684608 |
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| author | Rui Yang Mingzhu Yuan |
| author_facet | Rui Yang Mingzhu Yuan |
| author_sort | Rui Yang |
| collection | DOAJ |
| description | A <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene <i>F</i> is a 2-connected cubic planar graph whose faces are only <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn></mrow></semantics></math></inline-formula>-length and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>6</mn></mrow></semantics></math></inline-formula>-length. Furthermore, it consists of exactly three <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn></mrow></semantics></math></inline-formula>-length faces by Euler’s formula. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene comes from Došlić’s <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene, a 2-connected 3-regular plane graph with only <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi></mrow></semantics></math></inline-formula>-length faces and hexagons. Došlić showed that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerenes only exist for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, 3, 4, or 5, and some of the structural properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene. We discover that the edge-connectivity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerenes is 2. Every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene is 1-extendable, but not 2-extendable (<i>F</i> is called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mtext>-</mtext></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>e</mi><mi>x</mi><mi>t</mi><mi>e</mi><mi>n</mi><mi>d</mi><mi>a</mi><mi>b</mi><mi>l</mi><mi>e</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>V</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula>) if any matching of <i>n</i> edges is contained in a perfect matching of <i>F</i>). <i>F</i> is said to be <i>k</i>-<i>resonant</i> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>) if the deleting of any <i>i</i> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></semantics></math></inline-formula>) disjoint even faces of <i>F</i> results in a graph with at least one perfect matching. We have that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene is 1-resonant. An edge set, <i>S</i>, of <i>F</i> is called an anti−Kekulé set if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>−</mo><mi>S</mi></mrow></semantics></math></inline-formula> is connected and has no perfect matchings, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>−</mo><mi>S</mi></mrow></semantics></math></inline-formula> denotes the subgraph obtained by deleting all edges in <i>S</i> from <i>F</i>. The anti−Kekulé number of <i>F</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>k</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is the cardinality of a smallest anti−Kekulé set of <i>F</i>. We have that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene <i>F</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>V</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>|</mo><mo>></mo><mn>6</mn></mrow></semantics></math></inline-formula> has anti−Kekulé number 4. Further we mainly prove that there exists a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene <i>F</i> having <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mi>F</mi></msub></semantics></math></inline-formula> hexagonal faces, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mi>F</mi></msub></semantics></math></inline-formula> is related to the two parameters <i>n</i> and <i>m</i>. |
| first_indexed | 2024-03-09T16:24:45Z |
| format | Article |
| id | doaj.art-94df7d5d714c402e9a139bf45e35f99a |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T16:24:45Z |
| publishDate | 2023-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-94df7d5d714c402e9a139bf45e35f99a2023-11-24T15:09:00ZengMDPI AGSymmetry2073-89942023-11-011511207810.3390/sym15112078The Structural Properties of (2, 6)-FullerenesRui Yang0Mingzhu Yuan1School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, ChinaSchool of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, ChinaA <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene <i>F</i> is a 2-connected cubic planar graph whose faces are only <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn></mrow></semantics></math></inline-formula>-length and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>6</mn></mrow></semantics></math></inline-formula>-length. Furthermore, it consists of exactly three <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn></mrow></semantics></math></inline-formula>-length faces by Euler’s formula. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene comes from Došlić’s <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene, a 2-connected 3-regular plane graph with only <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi></mrow></semantics></math></inline-formula>-length faces and hexagons. Došlić showed that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerenes only exist for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, 3, 4, or 5, and some of the structural properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene. We discover that the edge-connectivity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerenes is 2. Every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene is 1-extendable, but not 2-extendable (<i>F</i> is called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mtext>-</mtext></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>e</mi><mi>x</mi><mi>t</mi><mi>e</mi><mi>n</mi><mi>d</mi><mi>a</mi><mi>b</mi><mi>l</mi><mi>e</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>V</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula>) if any matching of <i>n</i> edges is contained in a perfect matching of <i>F</i>). <i>F</i> is said to be <i>k</i>-<i>resonant</i> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>) if the deleting of any <i>i</i> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></semantics></math></inline-formula>) disjoint even faces of <i>F</i> results in a graph with at least one perfect matching. We have that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene is 1-resonant. An edge set, <i>S</i>, of <i>F</i> is called an anti−Kekulé set if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>−</mo><mi>S</mi></mrow></semantics></math></inline-formula> is connected and has no perfect matchings, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>−</mo><mi>S</mi></mrow></semantics></math></inline-formula> denotes the subgraph obtained by deleting all edges in <i>S</i> from <i>F</i>. The anti−Kekulé number of <i>F</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>k</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is the cardinality of a smallest anti−Kekulé set of <i>F</i>. We have that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene <i>F</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>V</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>|</mo><mo>></mo><mn>6</mn></mrow></semantics></math></inline-formula> has anti−Kekulé number 4. Further we mainly prove that there exists a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene <i>F</i> having <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mi>F</mi></msub></semantics></math></inline-formula> hexagonal faces, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mi>F</mi></msub></semantics></math></inline-formula> is related to the two parameters <i>n</i> and <i>m</i>.https://www.mdpi.com/2073-8994/15/11/2078(2, 6)-fullereneedge-connectivityanti-Kekulé numberresonance |
| spellingShingle | Rui Yang Mingzhu Yuan The Structural Properties of (2, 6)-Fullerenes Symmetry (2, 6)-fullerene edge-connectivity anti-Kekulé number resonance |
| title | The Structural Properties of (2, 6)-Fullerenes |
| title_full | The Structural Properties of (2, 6)-Fullerenes |
| title_fullStr | The Structural Properties of (2, 6)-Fullerenes |
| title_full_unstemmed | The Structural Properties of (2, 6)-Fullerenes |
| title_short | The Structural Properties of (2, 6)-Fullerenes |
| title_sort | structural properties of 2 6 fullerenes |
| topic | (2, 6)-fullerene edge-connectivity anti-Kekulé number resonance |
| url | https://www.mdpi.com/2073-8994/15/11/2078 |
| work_keys_str_mv | AT ruiyang thestructuralpropertiesof26fullerenes AT mingzhuyuan thestructuralpropertiesof26fullerenes AT ruiyang structuralpropertiesof26fullerenes AT mingzhuyuan structuralpropertiesof26fullerenes |