On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue
The signless Laplacian reciprocal distance matrix for a simple connected graph <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>Q</mi>...
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MDPI AG
2021-03-01
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| Online Access: | https://www.mdpi.com/2227-7390/9/5/512 |
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| author | Maryam Baghipur Modjtaba Ghorbani Hilal A. Ganie Yilun Shang |
| author_facet | Maryam Baghipur Modjtaba Ghorbani Hilal A. Ganie Yilun Shang |
| author_sort | Maryam Baghipur |
| collection | DOAJ |
| description | The signless Laplacian reciprocal distance matrix for a simple connected graph <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>diag</mi><mo>(</mo><mi>R</mi><mi>H</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>R</mi><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Here, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the Harary matrix (also called reciprocal distance matrix) while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>diag</mi><mo>(</mo><mi>R</mi><mi>H</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with <i>n</i> vertices, the complete graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>n</mi></msub></semantics></math></inline-formula> and the graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mi>n</mi></msub><mo>−</mo><mi>e</mi></mrow></semantics></math></inline-formula> obtained from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>n</mi></msub></semantics></math></inline-formula> by deleting an edge <i>e</i> have the maximum second-largest signless Laplacian reciprocal distance eigenvalue. |
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| last_indexed | 2024-03-09T06:00:24Z |
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| spelling | doaj.art-a8306292a31049ba85227c0623ab92c42023-12-03T12:09:19ZengMDPI AGMathematics2227-73902021-03-019551210.3390/math9050512On the Second-Largest Reciprocal Distance Signless Laplacian EigenvalueMaryam Baghipur0Modjtaba Ghorbani1Hilal A. Ganie2Yilun Shang3Department of Mathematics, Faculty of Science, Shahid Rajaee, Teacher Training University, Tehran 16785-136, IranDepartment of Mathematics, Faculty of Science, Shahid Rajaee, Teacher Training University, Tehran 16785-136, IranDepartment of School Education, Jammu and Kashmir Government, Kashmir 193404, IndiaDepartment of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UKThe signless Laplacian reciprocal distance matrix for a simple connected graph <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>diag</mi><mo>(</mo><mi>R</mi><mi>H</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>R</mi><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Here, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the Harary matrix (also called reciprocal distance matrix) while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>diag</mi><mo>(</mo><mi>R</mi><mi>H</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with <i>n</i> vertices, the complete graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>n</mi></msub></semantics></math></inline-formula> and the graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mi>n</mi></msub><mo>−</mo><mi>e</mi></mrow></semantics></math></inline-formula> obtained from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>n</mi></msub></semantics></math></inline-formula> by deleting an edge <i>e</i> have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.https://www.mdpi.com/2227-7390/9/5/512signless Laplacian reciprocal distance matrix (spectrum)spectral radiustotal reciprocal distance vertexHarary matrix |
| spellingShingle | Maryam Baghipur Modjtaba Ghorbani Hilal A. Ganie Yilun Shang On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue Mathematics signless Laplacian reciprocal distance matrix (spectrum) spectral radius total reciprocal distance vertex Harary matrix |
| title | On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue |
| title_full | On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue |
| title_fullStr | On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue |
| title_full_unstemmed | On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue |
| title_short | On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue |
| title_sort | on the second largest reciprocal distance signless laplacian eigenvalue |
| topic | signless Laplacian reciprocal distance matrix (spectrum) spectral radius total reciprocal distance vertex Harary matrix |
| url | https://www.mdpi.com/2227-7390/9/5/512 |
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