The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks
The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let Ln8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of Ln8,4, we get the corresponding Möbius graph MQn8,4. In this paper, starti...
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Wiley
2021-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/2328940 |
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| author | Jia-Bao Liu Qian Zheng Sakander Hayat |
| author_facet | Jia-Bao Liu Qian Zheng Sakander Hayat |
| author_sort | Jia-Bao Liu |
| collection | DOAJ |
| description | The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let Ln8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of Ln8,4, we get the corresponding Möbius graph MQn8,4. In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of MQn8,4 can be determined by the eigenvalues of two symmetric quasi-triangular matrices ℒA and ℒS of order 4n. Next, owing to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of MQn8,4. |
| format | Article |
| id | doaj-art-9273267f616344a5b997822c23771c9b |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-9273267f616344a5b997822c23771c9b2025-02-03T07:24:25ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/23289402328940The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral NetworksJia-Bao Liu0Qian Zheng1Sakander Hayat2School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, ChinaSchool of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, ChinaFaculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi Swabi, Khyber Pakhtunkhwa, PakistanThe normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let Ln8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of Ln8,4, we get the corresponding Möbius graph MQn8,4. In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of MQn8,4 can be determined by the eigenvalues of two symmetric quasi-triangular matrices ℒA and ℒS of order 4n. Next, owing to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of MQn8,4.http://dx.doi.org/10.1155/2021/2328940 |
| spellingShingle | Jia-Bao Liu Qian Zheng Sakander Hayat The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks Journal of Mathematics |
| title | The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks |
| title_full | The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks |
| title_fullStr | The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks |
| title_full_unstemmed | The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks |
| title_short | The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks |
| title_sort | normalized laplacians degree kirchhoff index and the complexity of mobius graph of linear octagonal quadrilateral networks |
| url | http://dx.doi.org/10.1155/2021/2328940 |
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