Authors: Alazemi, Abdullah
Anđelić, Milica
Simić, Slobodan 
Title: Eigenvalue location for chain graphs
Journal: Linear Algebra and Its Applications
Volume: 505
First page: 194
Last page: 210
Issue Date: 15-Sep-2016
Rank: M21
ISSN: 0024-3795
DOI: 10.1016/j.laa.2016.04.030
Chain graphs (also called double nested graphs) play an important role in the spectral graph theory since every connected bipartite graph of fixed order and size with maximal largest eigenvalue is a chain graph. In this paper, for a given chain graph G, we present an algorithmic procedure for obtaining a diagonal matrix congruent to A+xI, where A is the adjacency matrix of G and x any real number. Using this procedure we show that any chain graph has its least positive eigenvalue greater than 12, and also prove that this bound is best possible. A similar procedure for threshold graphs (also called nested split graphs) is outlined.
Keywords: Chain graph | Double nested graph | Least positive eigenvalue | Nested split graph | Threshold graph
Publisher: Elsevier
Project: Kuwait University, Grant No. SM03/15

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