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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
DOI: 10.1016/j.laa.2015.01.026
ISSN: 0024-3795
Appears in Collections:CIDMA - Artigos
OGTCG - Artigos

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