Extremal Results on <i>ℓ</i>-Connected Graphs or Pancyclic Graphs Based on Wiener-Type Indices
A graph of order <i>n</i> is called pancyclic if it contains a cycle of length <i>y</i> for every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>≤&l...
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| author | Jing Zeng Hechao Liu Lihua You |
| author_facet | Jing Zeng Hechao Liu Lihua You |
| author_sort | Jing Zeng |
| collection | DOAJ |
| description | A graph of order <i>n</i> is called pancyclic if it contains a cycle of length <i>y</i> for every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>. The connectivity of an incomplete graph <i>G</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">min</mo><mo>{</mo><mo>|</mo><mi>W</mi><mo>|</mo><mo>|</mo><mi>W</mi><mspace width="4pt"></mspace><mi>i</mi><mi>s</mi><mspace width="4pt"></mspace><mi>a</mi><mspace width="4pt"></mspace><mi>v</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>x</mi><mspace width="4pt"></mspace><mi>c</mi><mi>u</mi><mi>t</mi><mspace width="4pt"></mspace><mi>o</mi><mi>f</mi><mspace width="4pt"></mspace><mi>G</mi><mo>}</mo></mrow></semantics></math></inline-formula>. A graph <i>G</i> is said to be <i>ℓ</i>-connected if the connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mo>ℓ</mo></mrow></semantics></math></inline-formula>. The Wiener-type indices of a connected graph <i>G</i> are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mi>g</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mo>{</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>}</mo><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munder></mstyle><mi>g</mi><mrow><mo>(</mo><msub><mi>d</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a function and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the distance in <i>G</i> between <i>s</i> and <i>t</i>. In this note, we first determine the minimum and maximum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mi>g</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <i>ℓ</i>-connected graphs. Then, we use the Wiener-type indices of graph <i>G</i>, the Wiener-type indices of complement graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>G</mi><mo>¯</mo></mover></semantics></math></inline-formula> with minimum degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula> to give some sufficient conditions for connected graphs to be pancyclic. Our results generalize some existing results of several papers. |
| format | Article |
| id | doaj-art-4a4afa9c4d174428b6f885d468cf1c90 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-4a4afa9c4d174428b6f885d468cf1c902025-01-10T13:17:57ZengMDPI AGMathematics2227-73902024-12-011311010.3390/math13010010Extremal Results on <i>ℓ</i>-Connected Graphs or Pancyclic Graphs Based on Wiener-Type IndicesJing Zeng0Hechao Liu1Lihua You2School of Mathematical Sciences, South China Normal University, Guangzhou 510631, ChinaSchool of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, ChinaSchool of Mathematical Sciences, South China Normal University, Guangzhou 510631, ChinaA graph of order <i>n</i> is called pancyclic if it contains a cycle of length <i>y</i> for every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>. The connectivity of an incomplete graph <i>G</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">min</mo><mo>{</mo><mo>|</mo><mi>W</mi><mo>|</mo><mo>|</mo><mi>W</mi><mspace width="4pt"></mspace><mi>i</mi><mi>s</mi><mspace width="4pt"></mspace><mi>a</mi><mspace width="4pt"></mspace><mi>v</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>x</mi><mspace width="4pt"></mspace><mi>c</mi><mi>u</mi><mi>t</mi><mspace width="4pt"></mspace><mi>o</mi><mi>f</mi><mspace width="4pt"></mspace><mi>G</mi><mo>}</mo></mrow></semantics></math></inline-formula>. A graph <i>G</i> is said to be <i>ℓ</i>-connected if the connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mo>ℓ</mo></mrow></semantics></math></inline-formula>. The Wiener-type indices of a connected graph <i>G</i> are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mi>g</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mo>{</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>}</mo><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munder></mstyle><mi>g</mi><mrow><mo>(</mo><msub><mi>d</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a function and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the distance in <i>G</i> between <i>s</i> and <i>t</i>. In this note, we first determine the minimum and maximum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mi>g</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <i>ℓ</i>-connected graphs. Then, we use the Wiener-type indices of graph <i>G</i>, the Wiener-type indices of complement graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>G</mi><mo>¯</mo></mover></semantics></math></inline-formula> with minimum degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula> to give some sufficient conditions for connected graphs to be pancyclic. Our results generalize some existing results of several papers.https://www.mdpi.com/2227-7390/13/1/10ℓ-connected graphpancyclic graphWiener-type indexsufficient condition |
| spellingShingle | Jing Zeng Hechao Liu Lihua You Extremal Results on <i>ℓ</i>-Connected Graphs or Pancyclic Graphs Based on Wiener-Type Indices Mathematics ℓ-connected graph pancyclic graph Wiener-type index sufficient condition |
| title | Extremal Results on <i>ℓ</i>-Connected Graphs or Pancyclic Graphs Based on Wiener-Type Indices |
| title_full | Extremal Results on <i>ℓ</i>-Connected Graphs or Pancyclic Graphs Based on Wiener-Type Indices |
| title_fullStr | Extremal Results on <i>ℓ</i>-Connected Graphs or Pancyclic Graphs Based on Wiener-Type Indices |
| title_full_unstemmed | Extremal Results on <i>ℓ</i>-Connected Graphs or Pancyclic Graphs Based on Wiener-Type Indices |
| title_short | Extremal Results on <i>ℓ</i>-Connected Graphs or Pancyclic Graphs Based on Wiener-Type Indices |
| title_sort | extremal results on i l i connected graphs or pancyclic graphs based on wiener type indices |
| topic | ℓ-connected graph pancyclic graph Wiener-type index sufficient condition |
| url | https://www.mdpi.com/2227-7390/13/1/10 |
| work_keys_str_mv | AT jingzeng extremalresultsoniliconnectedgraphsorpancyclicgraphsbasedonwienertypeindices AT hechaoliu extremalresultsoniliconnectedgraphsorpancyclicgraphsbasedonwienertypeindices AT lihuayou extremalresultsoniliconnectedgraphsorpancyclicgraphsbasedonwienertypeindices |