On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

For a graph \(G\), the unraveled ball of radius \(r\) centered at a vertex \(v\) is the ball of radius \(r\) centered at \(v\) in the universal cover of \(G\). We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which...

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Bibliographic Details
Published inarXiv.org
Main Authors Wang, Yuzhenni, Xiao-Dong, Zhang
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 22.09.2022
Subjects
Eigenvalues
Graph theory
Graphs
Lower bounds
Upper bounds
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Summary:For a graph \(G\), the unraveled ball of radius \(r\) centered at a vertex \(v\) is the ball of radius \(r\) centered at \(v\) in the universal cover of \(G\). We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the \(s\)-th (where \(s\ge 2\)) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when \(s=2\), the result may be regarded as an Alon--Boppana type bound for a class of irregular graphs.
ISSN:2331-8422