Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs
The Laplacian spectra are the eigenvalues of Laplacian matrix L(G)=D(G)-A(G), where D(G) and A(G) are the diagonal matrix of vertex degrees and the adjacency matrix of a graph G, respectively, and the spectral radius of a graph G is the largest eigenvalue of A(G). The spectra of the graph and corres...
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Wiley
2013-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2013/472956 |
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| author | Tianfei Wang Liping Jia Feng Sun |
| author_facet | Tianfei Wang Liping Jia Feng Sun |
| author_sort | Tianfei Wang |
| collection | DOAJ |
| description | The Laplacian spectra are the eigenvalues of Laplacian matrix L(G)=D(G)-A(G), where D(G) and A(G) are the diagonal matrix of vertex degrees and the adjacency matrix of a graph G, respectively, and the spectral radius of a graph G is the largest eigenvalue of A(G). The spectra of the graph and corresponding eigenvalues are closely linked to the molecular stability and related chemical properties. In quantum chemistry, spectral radius of a graph is the maximum energy level of molecules. Therefore, good upper bounds for the spectral radius are conducive to evaluate the energy of molecules. In this paper, we first give several sharp upper bounds on the adjacency spectral radius in terms of some invariants of graphs, such as the vertex degree, the average 2-degree, and the number of the triangles. Then, we give some numerical examples which indicate that the results are better than the mentioned upper bounds in some sense. Finally, an upper bound of the Nordhaus-Gaddum type is obtained for the sum of Laplacian spectral radius of a connected graph and its complement. Moreover, some examples are applied to illustrate that our result is valuable. |
| format | Article |
| id | doaj-art-1782a23761324f31be6ec1f1d0386588 |
| institution | Kabale University |
| issn | 1537-744X |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-1782a23761324f31be6ec1f1d03865882025-02-03T06:01:37ZengWileyThe Scientific World Journal1537-744X2013-01-01201310.1155/2013/472956472956Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the GraphsTianfei Wang0Liping Jia1Feng Sun2School of Mathematics and Information Science, Leshan Normal University, Leshan 614004, ChinaSchool of Mathematics and Information Science, Leshan Normal University, Leshan 614004, ChinaSchool of Mathematics and Information Science, Leshan Normal University, Leshan 614004, ChinaThe Laplacian spectra are the eigenvalues of Laplacian matrix L(G)=D(G)-A(G), where D(G) and A(G) are the diagonal matrix of vertex degrees and the adjacency matrix of a graph G, respectively, and the spectral radius of a graph G is the largest eigenvalue of A(G). The spectra of the graph and corresponding eigenvalues are closely linked to the molecular stability and related chemical properties. In quantum chemistry, spectral radius of a graph is the maximum energy level of molecules. Therefore, good upper bounds for the spectral radius are conducive to evaluate the energy of molecules. In this paper, we first give several sharp upper bounds on the adjacency spectral radius in terms of some invariants of graphs, such as the vertex degree, the average 2-degree, and the number of the triangles. Then, we give some numerical examples which indicate that the results are better than the mentioned upper bounds in some sense. Finally, an upper bound of the Nordhaus-Gaddum type is obtained for the sum of Laplacian spectral radius of a connected graph and its complement. Moreover, some examples are applied to illustrate that our result is valuable.http://dx.doi.org/10.1155/2013/472956 |
| spellingShingle | Tianfei Wang Liping Jia Feng Sun Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs The Scientific World Journal |
| title | Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs |
| title_full | Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs |
| title_fullStr | Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs |
| title_full_unstemmed | Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs |
| title_short | Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs |
| title_sort | bounds of the spectral radius and the nordhaus gaddum type of the graphs |
| url | http://dx.doi.org/10.1155/2013/472956 |
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