On the Generalized Adjacency Spread of a Graph
For a simple finite graph <i>G</i>, the generalized adjacency matrix is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">A</mi&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/6/1416 |
| _version_ | 1827748921950076928 |
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| author | Maryam Baghipur Modjtaba Ghorbani Shariefuddin Pirzada Najaf Amraei |
| author_facet | Maryam Baghipur Modjtaba Ghorbani Shariefuddin Pirzada Najaf Amraei |
| author_sort | Maryam Baghipur |
| collection | DOAJ |
| description | For a simple finite graph <i>G</i>, the generalized adjacency matrix is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><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>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> are respectively the adjacency matrix and diagonal matrix of the vertex degrees. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula>-spread of a graph <i>G</i> is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we answer the question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula>-spread of a graph. <i>Electron. J. Linear Algebra </i><b>2020</b>, <i>36</i>, 214–227). Furthermore, we show that the path graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula>, has the smallest <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> among all trees of order <i>n</i>. We establish a relationship between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> We obtain several bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-11T06:13:43Z |
| format | Article |
| id | doaj.art-c98c897b4e2f46bc8f2a40ef885d3ee7 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T06:13:43Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c98c897b4e2f46bc8f2a40ef885d3ee72023-11-17T12:28:16ZengMDPI AGMathematics2227-73902023-03-01116141610.3390/math11061416On the Generalized Adjacency Spread of a GraphMaryam Baghipur0Modjtaba Ghorbani1Shariefuddin Pirzada2Najaf Amraei3Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-163, IranDepartment of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-163, IranDepartment of Mathematics, University of Kashmir, Srinagar 192101, IndiaDepartment of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-163, IranFor a simple finite graph <i>G</i>, the generalized adjacency matrix is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><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>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> are respectively the adjacency matrix and diagonal matrix of the vertex degrees. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula>-spread of a graph <i>G</i> is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we answer the question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula>-spread of a graph. <i>Electron. J. Linear Algebra </i><b>2020</b>, <i>36</i>, 214–227). Furthermore, we show that the path graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula>, has the smallest <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> among all trees of order <i>n</i>. We establish a relationship between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> We obtain several bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/11/6/1416generalized adjacency matrixspreadeigenvalue |
| spellingShingle | Maryam Baghipur Modjtaba Ghorbani Shariefuddin Pirzada Najaf Amraei On the Generalized Adjacency Spread of a Graph Mathematics generalized adjacency matrix spread eigenvalue |
| title | On the Generalized Adjacency Spread of a Graph |
| title_full | On the Generalized Adjacency Spread of a Graph |
| title_fullStr | On the Generalized Adjacency Spread of a Graph |
| title_full_unstemmed | On the Generalized Adjacency Spread of a Graph |
| title_short | On the Generalized Adjacency Spread of a Graph |
| title_sort | on the generalized adjacency spread of a graph |
| topic | generalized adjacency matrix spread eigenvalue |
| url | https://www.mdpi.com/2227-7390/11/6/1416 |
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