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DC Field | Value | Language |
---|---|---|

dc.contributor.author | Cardoso, Domingos M. | pt |

dc.contributor.author | Pastén, Germain | pt |

dc.contributor.author | Rojo, Oscar | pt |

dc.date.accessioned | 2018-04-30T10:04:19Z | - |

dc.date.issued | 2018-04-16 | - |

dc.identifier.issn | 0024-3795 | pt |

dc.identifier.uri | http://hdl.handle.net/10773/23003 | - |

dc.description.abstract | Let $G$ be a simple undirected graph. Let $0\leq \alpha \leq 1$. Let $$A_{\alpha}(G)= \alpha D(G) + (1-\alpha) A(G)$$ where $D(G)$ and $A(G)$ are the diagonal matrix of the vertex degrees of $G$ and the adjacency matrix of $G$, respectively. Let $p(G)>0$ and $q(G)$ be the number of pendant vertices and quasi-pendant vertices of $G$, respectively. Let $m_{G}(\alpha)$ be the multiplicity of $\alpha$ as eigenvalue of $A_{\alpha}(G)$. It is proved that \begin{equation*} m_{G}(\alpha) \geq p(G) - q(G) \end{equation*} with equality if each internal vertex is a quasi-pendant vertex. If there is at least one internal vertex which is not a quasi-pendant vertex, the equality \begin{equation*} m_{G}(\alpha)= p(G)-q(G)+m_{N}(\alpha) \end{equation*} is determined in which $m_{N}(\alpha)$ is the multiplicity of $\alpha$ as eigenvalue of the matrix $N$. This matrix is obtained from $A_{\alpha}(G)$ taking the entries corresponding to the internal vertices which are non quasi-pendant vertices. These results are applied to search for the multiplicity of $\alpha$ as eigenvalue of $A_{\alpha}(G)$ when $G$ is a path, a caterpillar, a circular caterpillar, a generalized Bethe tree or a Bethe tree. For the Bethe tree case, a simple formula for the nullity is given. | pt |

dc.language.iso | eng | pt |

dc.publisher | Elsevier | pt |

dc.relation | info:eu-repo/grantAgreement/FCT/5876/147206/PT | pt |

dc.relation | CONICYT-PCHA/MagisterNacional/2015-22151590 | pt |

dc.rights | openAccess | por |

dc.subject | Adjacency matrix | pt |

dc.subject | Signless Laplacian matrix | pt |

dc.subject | Laplacian matrix | pt |

dc.subject | Convex combination of matrices | pt |

dc.subject | Graph eigenvalues | pt |

dc.title | On the multiplicity of α as an eigenvalue of Aα(G) of graphs with pendant vertices | pt |

dc.type | article | pt |

dc.peerreviewed | yes | pt |

ua.distribution | international | pt |

degois.publication.firstPage | 52 | pt |

degois.publication.lastPage | 70 | pt |

degois.publication.title | Linear Algebra and Its Applications | pt |

degois.publication.volume | 552 | pt |

dc.date.embargo | 2020-04-09T10:00:00Z | - |

dc.identifier.doi | 10.1016/j.laa.2018.04.013 | pt |

Appears in Collections: | CIDMA - Artigos OGTCG - Artigos |

Files in This Item:

File | Description | Size | Format | |
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1-s2.0-S002437951830199X-main.pdf | Accepted Maniscript | 316.14 kB | Adobe PDF | View/Open |

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