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The spectral radius of submatrices of Laplacian matrices for graphs with cut vertices

โœ Scribed by Jason J. Molitierno

Publisher
Elsevier Science
Year
2008
Tongue
English
Weight
326 KB
Volume
428
Category
Article
ISSN
0024-3795

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โœฆ Synopsis

In [J. Molitierno, The spectral radius of submatrices of Laplacian matrices for trees and its comparison to the Fiedler vector, Linear Algebra Appl. 406 (2005) 253-271], we observed the effects on the spectral radius of submatrices of the Laplacian matrix L for a tree by deleting a row and column of L corresponding to a vertex of the tree. This enabled us to classify trees as either of Type A or Type B. In this paper, we extend these results to graphs which are not trees and offer a similar classification. Additionally, we show counterexamples to theorems that are true for trees, but not so for general graphs.

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We consider the effects on the spectral radius of submatrices of the Laplacian matrix for graphs by deleting the row and column corresponding to various vertices of the graph. We focus most of our attention on trees and determine which vertices v will yield the maximum and minimum spectral radius of