Please use this identifier to cite or link to this item: http://hdl.handle.net/10773/23003
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dc.contributor.authorCardoso, Domingos M.pt
dc.contributor.authorPastén, Germainpt
dc.contributor.authorRojo, Oscarpt
dc.date.accessioned2018-04-30T10:04:19Z-
dc.date.issued2018-04-16-
dc.identifier.issn0024-3795pt
dc.identifier.urihttp://hdl.handle.net/10773/23003-
dc.description.abstractLet $G$ 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.isoengpt
dc.publisherElsevierpt
dc.relationinfo:eu-repo/grantAgreement/FCT/5876/147206/PTpt
dc.relationCONICYT-PCHA/MagisterNacional/2015-22151590pt
dc.rightsopenAccesspor
dc.subjectAdjacency matrixpt
dc.subjectSignless Laplacian matrixpt
dc.subjectLaplacian matrixpt
dc.subjectConvex combination of matricespt
dc.subjectGraph eigenvaluespt
dc.titleOn the multiplicity of α as an eigenvalue of Aα(G) of graphs with pendant verticespt
dc.typearticlept
dc.peerreviewedyespt
ua.distributioninternationalpt
degois.publication.firstPage52pt
degois.publication.lastPage70pt
degois.publication.titleLinear Algebra and Its Applicationspt
degois.publication.volume552pt
dc.date.embargo2020-04-09T10:00:00Z-
dc.identifier.doi10.1016/j.laa.2018.04.013pt
Appears in Collections:CIDMA - Artigos
OGTCG - Artigos

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