Wiener–Hosoya Matrix of Connected Graphs
Let <i>G</i> be a connected (molecular) graph with the vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi>&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/4/359 |
| _version_ | 1797411414851190784 |
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| author | Hassan Ibrahim Reza Sharafdini Tamás Réti Abolape Akwu |
| author_facet | Hassan Ibrahim Reza Sharafdini Tamás Réti Abolape Akwu |
| author_sort | Hassan Ibrahim |
| collection | DOAJ |
| description | Let <i>G</i> be a connected (molecular) graph with the vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>v</mi><mi>n</mi></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>i</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mi>i</mi></msub></semantics></math></inline-formula> denote, respectively, the vertex degree and the transmission of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of <i>G</i>, which is defined as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> matrix whose <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></semantics></math></inline-formula>-entry is equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><msub><mi>σ</mi><mi>i</mi></msub><mrow><mn>2</mn><msub><mi>d</mi><mi>i</mi></msub></mrow></mfrac><mo>+</mo><mfrac><msub><mi>σ</mi><mi>j</mi></msub><mrow><mn>2</mn><msub><mi>d</mi><mi>j</mi></msub></mrow></mfrac></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>j</mi></msub></semantics></math></inline-formula> are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix. |
| first_indexed | 2024-03-09T04:45:48Z |
| format | Article |
| id | doaj.art-433a7d86d21f4153bfcedbadb2ee9f5c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T04:45:48Z |
| publishDate | 2021-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-433a7d86d21f4153bfcedbadb2ee9f5c2023-12-03T13:16:05ZengMDPI AGMathematics2227-73902021-02-019435910.3390/math9040359Wiener–Hosoya Matrix of Connected GraphsHassan Ibrahim0Reza Sharafdini1Tamás Réti2Abolape Akwu3Department of Mathematics Statistics & Computer Science, Federal University of Agriculture, Makurdi P.M.B 2373, NigeriaDepartment of Mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr 75169, IranBánki Donát Faculty of Mechanical and Safety Engineering, Óbuda University Bécsiút 96/B, H-1034 Budapest, HungaryDepartment of Mathematics Statistics & Computer Science, Federal University of Agriculture, Makurdi P.M.B 2373, NigeriaLet <i>G</i> be a connected (molecular) graph with the vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>v</mi><mi>n</mi></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>i</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mi>i</mi></msub></semantics></math></inline-formula> denote, respectively, the vertex degree and the transmission of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of <i>G</i>, which is defined as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> matrix whose <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></semantics></math></inline-formula>-entry is equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><msub><mi>σ</mi><mi>i</mi></msub><mrow><mn>2</mn><msub><mi>d</mi><mi>i</mi></msub></mrow></mfrac><mo>+</mo><mfrac><msub><mi>σ</mi><mi>j</mi></msub><mrow><mn>2</mn><msub><mi>d</mi><mi>j</mi></msub></mrow></mfrac></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>j</mi></msub></semantics></math></inline-formula> are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.https://www.mdpi.com/2227-7390/9/4/359transmissionvertex-degreeWiener indexspectral radiusenergy |
| spellingShingle | Hassan Ibrahim Reza Sharafdini Tamás Réti Abolape Akwu Wiener–Hosoya Matrix of Connected Graphs Mathematics transmission vertex-degree Wiener index spectral radius energy |
| title | Wiener–Hosoya Matrix of Connected Graphs |
| title_full | Wiener–Hosoya Matrix of Connected Graphs |
| title_fullStr | Wiener–Hosoya Matrix of Connected Graphs |
| title_full_unstemmed | Wiener–Hosoya Matrix of Connected Graphs |
| title_short | Wiener–Hosoya Matrix of Connected Graphs |
| title_sort | wiener hosoya matrix of connected graphs |
| topic | transmission vertex-degree Wiener index spectral radius energy |
| url | https://www.mdpi.com/2227-7390/9/4/359 |
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