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Title: | On the Faria's inequality for the Laplacian and signless Laplacian spectra: a unified approach |

Author: | Andrade, E. Cardoso, Domingos M. Pastén, G. Rojo, O. |

Keywords: | Spectral graph theory Laplacian spectrum of a graph Signless Laplacian spectrum of a graph |

Issue Date: | 2015 |

Publisher: | Elsevier |

Abstract: | Let p(G)p(G) and q(G)q(G) be the number of pendant vertices and quasi-pendant vertices of a simple undirected graph G, respectively. Let m_L±(G)(1) be the multiplicity of 1 as eigenvalue of a matrix which can be either the Laplacian or the signless Laplacian of a graph G. A result due to I. Faria states that mL±(G)(1) is bounded below by p(G)−q(G). Let r(G) be the number of internal vertices of G. If r(G)=q(G), following a unified approach we prove that mL±(G)(1)=p(G)−q(G). If r(G)>q(G) then we determine the equality mL±(G)(1)=p(G)−q(G)+mN±(1), where mN±(1) denotes the multiplicity of 1 as eigenvalue of a matrix N±. This matrix is obtained from either the Laplacian or signless Laplacian matrix of the subgraph induced by the internal vertices which are non-quasi-pendant vertices. Furthermore, conditions for 1 to be an eigenvalue of a principal submatrix are deduced and applied to some families of graphs. |

Peer review: | yes |

URI: | http://hdl.handle.net/10773/13475 |

DOI: | 10.1016/j.laa.2015.01.026 |

ISSN: | 0024-3795 |

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

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1-s2.0-S0024379515000555-main.pdf | Research article | 354.07 kB | Adobe PDF | Request a copy |

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