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Title: | On the multiplicity of α as an eigenvalue of Aα(G) of graphs with pendant vertices |

Author: | Cardoso, Domingos M. Pastén, Germain Rojo, Oscar |

Keywords: | Adjacency matrix Signless Laplacian matrix Laplacian matrix Convex combination of matrices Graph eigenvalues |

Issue Date: | 16-Apr-2018 |

Publisher: | Elsevier |

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. |

Peer review: | yes |

URI: | http://hdl.handle.net/10773/23003 |

DOI: | 10.1016/j.laa.2018.04.013 |

ISSN: | 0024-3795 |

Appears in Collections: | CIDMA - Artigos |

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1-s2.0-S002437951830199X-main.pdf | Accepted Maniscript | 316.14 kB | Adobe PDF | Request a copy |

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