Theorems on partitioned matrices revisited and their applications to graph spectra

doi:10.1016/j.laa.2010.09.014

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Title:	Theorems on partitioned matrices revisited and their applications to graph spectra
Authors:	Chang, Ting-Chung;Tam, Bit-Shun;Wu, Shu-Hui
Contributors:	淡江大學數學學系
Keywords:	Graph spectra;Neighborhood equivalence class;Block-stochastic matrix;Laplacian;Signless laplacian
Date:	2010-10
Issue Date:	2013-06-13 11:24:37 (UTC+8)
Publisher:	Philadelphia: Elsevier Inc.
Abstract:	Some old results about spectra of partitioned matrices due to Goddard and Schneider or Haynsworth are re-proved. A new result is given for the spectrum of a block-stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. The result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation. The concept of a reduced graph matrix is introduced. The question of when n-2 is the second largest signless Laplacian eigenvalue of a connected graph of order n is treated. A recent conjecture posed by Tam, Fan and Zhou on graphs that maximize the signless Laplacian spectral radius over all (not necessarily connected) graphs with given numbers of vertices and edges is refuted. The Laplacian spectrum of a (degree) maximal graph is reconsidered.
Relation:	Linear Algebra and Its Applications 434(2), pp.559-581
DOI:	10.1016/j.laa.2010.09.014
Appears in Collections:	[Graduate Institute & Department of Mathematics] Journal Article

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