On the spectrum of an equitable quotient matrix and its application

Let M be a partitioned matrix and B be its equitable quotient matrix. In this paper, we prove the inclusion of their spectra σ(B)⊆σ(M), and the equality of their spectral radii ρ(B)=ρ(M) if M is nonnegative. Moreover, σ(M) is shown to be completely determined by σ(B) if M is from a class of special...

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Published inLinear algebra and its applications Vol. 577; pp. 21 - 40
Main Authors You, Lihua, Yang, Man, So, Wasin, Xi, Weige
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 15.09.2019
American Elsevier Company, Inc
Subjects
Digraph
Graph
Graph theory
Graphs
Linear algebra
Matrix
Quotient matrix
Quotients
Spectra
Spectral radius
Spectrum
05C20
Quotient matrix
Matrix
Graph
05C50
Spectral radius
Digraph
05C35
15A18
Spectrum
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Summary:Let M be a partitioned matrix and B be its equitable quotient matrix. In this paper, we prove the inclusion of their spectra σ(B)⊆σ(M), and the equality of their spectral radii ρ(B)=ρ(M) if M is nonnegative. Moreover, σ(M) is shown to be completely determined by σ(B) if M is from a class of special matrices. Then we apply these results to obtain the spectra of some well-known matrices associated with graphs or digraphs, and then determine the extremal values of different spectral radii associated with connected graphs or strongly connected digraphs with given vertex connectivity.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2019.04.013