Authors: Anđelić, Milica
Andrade, Enide
Cardoso, Domingos
Da Fonseca, Carlos
Simić, Slobodan 
Tošić, Dejan
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Some new considerations about double nested graphs
Journal: Linear Algebra and Its Applications
Volume: 483
First page: 323
Last page: 341
Issue Date: 2-Jul-2015
Rank: M21
ISSN: 0024-3795
DOI: 10.1016/j.laa.2015.06.010
In the set of all connected graphs with fixed order and size, the graphs with maximal index are nested split graphs, also called threshold graphs. It was recently (and independently) observed in Bell et al. (2008) [3] and Bhattacharya et al. (2008) [4] that double nested graphs, also called bipartite chain graphs, play the same role within class of bipartite graphs. In this paper we study some structural and spectral features of double nested graphs. In studying the spectrum of double nested graphs we rather consider some weighted nonnegative matrices (of significantly less order) which preserve all positive eigenvalues of former ones. Moreover, their inverse matrices appear to be tridiagonal. Using this fact we provide several new bounds on the index (largest eigenvalue) of double nested graphs, and also deduce some bounds on eigenvector components for the index. We conclude the paper by examining the questions related to main versus non-main eigenvalues.
Keywords: Bipartite graph | Double nested graph | Largest eigenvalue | Main eigenvalue | Spectral bounds
Publisher: Elsevier
Project: QREN project Cloud Thinking (CENTRO-07-ST24-FEDER-002031)
FCT - Fundação para a Ciência e a Tecnologia, Project PEst-UID/MAT/04106/2013
Graph theory and mathematical programming with applications in chemistry and computer science 
Development of new information and communication technologies, based on advanced mathematical methods, with applications in medicine, telecommunications, power systems, protection of national heritage and education 

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