On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent Vertices
Let Φ(G,λ)=det(λIn-L(G))=∑k=0n(-1)kck(G)λn-k be the characteristic polynomial of the Laplacian matrix of a graph G of order n. In this paper, we give four transforms on graphs that decrease all Laplacian coefficients ck(G) and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian c...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/404067 |
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| _version_ | 1832552378853752832 |
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| author | Xinying Pai Sanyang Liu |
| author_facet | Xinying Pai Sanyang Liu |
| author_sort | Xinying Pai |
| collection | DOAJ |
| description | Let Φ(G,λ)=det(λIn-L(G))=∑k=0n(-1)kck(G)λn-k be the characteristic polynomial of the Laplacian matrix of a graph G of order n. In this paper, we give four transforms on graphs that decrease all Laplacian coefficients ck(G) and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs with n vertices and m pendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with n vertices and m pendent vertices. |
| format | Article |
| id | doaj-art-a5755d7ea18943f08924a2c6012f61d1 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-a5755d7ea18943f08924a2c6012f61d12025-02-03T05:58:51ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/404067404067On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent VerticesXinying Pai0Sanyang Liu1Department of Mathematics, Xidian University, Shanxi Xi'an 710071, ChinaDepartment of Mathematics, Xidian University, Shanxi Xi'an 710071, ChinaLet Φ(G,λ)=det(λIn-L(G))=∑k=0n(-1)kck(G)λn-k be the characteristic polynomial of the Laplacian matrix of a graph G of order n. In this paper, we give four transforms on graphs that decrease all Laplacian coefficients ck(G) and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs with n vertices and m pendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with n vertices and m pendent vertices.http://dx.doi.org/10.1155/2012/404067 |
| spellingShingle | Xinying Pai Sanyang Liu On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent Vertices Journal of Applied Mathematics |
| title | On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent Vertices |
| title_full | On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent Vertices |
| title_fullStr | On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent Vertices |
| title_full_unstemmed | On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent Vertices |
| title_short | On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent Vertices |
| title_sort | on the laplacian coefficients and laplacian like energy of unicyclic graphs with n vertices and m pendent vertices |
| url | http://dx.doi.org/10.1155/2012/404067 |
| work_keys_str_mv | AT xinyingpai onthelaplaciancoefficientsandlaplacianlikeenergyofunicyclicgraphswithnverticesandmpendentvertices AT sanyangliu onthelaplaciancoefficientsandlaplacianlikeenergyofunicyclicgraphswithnverticesandmpendentvertices |