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Locating Eigenvalues in Graphs

Algorithms and Applications

  • Book
  • © 2022

Overview

Authors:
  1. Carlos Hoppen

    1. Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil

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  2. David P. Jacobs

    1. School of Computing, Clemson University, Clemson, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Vilmar Trevisan

    1. Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil

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  • Offers a careful exposition of a class of important eigenvalue location algorithms for various graph classes
  • Introduces spectral graph theory and graph representations concisely
  • Describes applications of location algorithms in spectral graph theory and presents open problems in the area

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • Carlos Hoppen, David P. Jacobs, Vilmar Trevisan
    Pages 1-4

  3. Preliminaries

    • Carlos Hoppen, David P. Jacobs, Vilmar Trevisan
    Pages 5-23

  4. Locating Eigenvalues in Trees

    • Carlos Hoppen, David P. Jacobs, Vilmar Trevisan
    Pages 25-37

  5. Graph Classes and Graph Decompositions

    • Carlos Hoppen, David P. Jacobs, Vilmar Trevisan
    Pages 39-63

  6. Locating Eigenvalues in Cographs

    • Carlos Hoppen, David P. Jacobs, Vilmar Trevisan
    Pages 65-79

  7. Locating Eigenvalues Using Tree Decomposition

    • Carlos Hoppen, David P. Jacobs, Vilmar Trevisan
    Pages 81-101

  8. Locating Eigenvalues Using Slick Clique Decomposition

    • Carlos Hoppen, David P. Jacobs, Vilmar Trevisan
    Pages 103-120

  9. Distance-Hereditary Graphs

    • Carlos Hoppen, David P. Jacobs, Vilmar Trevisan
    Pages 121-132

  10. Back Matter

    Pages 133-136
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Keywords

About this book

This book focuses on linear time eigenvalue location algorithms for graphs. This subject relates to spectral graph theory, a field that combines tools and concepts of linear algebra and combinatorics, with applications ranging from image processing and data analysis to molecular descriptors and random walks. It has attracted a lot of attention and has since emerged as an area on its own.


Studies in spectral graph theory seek to determine properties of a graph through matrices associated with it. It turns out that eigenvalues and eigenvectors have surprisingly many connections with the structure of a graph. This book approaches this subject under the perspective of eigenvalue location algorithms. These are algorithms that, given a symmetric graph matrix M and a real interval I, return the number of eigenvalues of M that lie in I. Since the algorithms described here are typically very fast, they allow one to quickly approximate the value of any eigenvalue, which is a basic step in most applications of spectral graph theory. Moreover, these algorithms are convenient theoretical tools for proving bounds on eigenvalues and their multiplicities, which was quite useful to solve longstanding open problems in the area. This book brings these algorithms together, revealing how similar they are in spirit, and presents some of their main applications.


This work can be of special interest to graduate students and researchers in spectral graph theory, and to any mathematician who wishes to know more about eigenvalues associated with graphs. It can also serve as a compact textbook for short courses on the topic.



Reviews

“The book is well structured, and gives the necessary background, detailed insight into eigenvalue location algorithms as well as their applications and possible relations with some open problems. It provides a concise and at the same time rich introduction to eigenvalue location algorithms.” (Milica Andelić, zbMATH 1511.05001, 2023)

Authors and Affiliations

  • Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil

    Carlos Hoppen, Vilmar Trevisan

  • School of Computing, Clemson University, Clemson, USA

    David P. Jacobs

About the authors

Carlos Hoppen holds a PhD in Combinatorics and Optimization from the University of Waterloo, Canada. He is an Associate Professor of Mathematics at the Federal University of Rio Grande do Sul (UFRGS), Brazil. His research focuses on probabilistic and extremal combinatorics, and spectral graph theory.


David P. Jacobs is a Professor Emeritus of Computer Science at Clemson University, USA. He has done research in various areas including graph algorithms and spectral graph theory. In 2006, he visited UFRGS as a Fulbright Scholar.


Vilmar Trevisan is a Professor of Mathematics at UFRGS. He earned a PhD in Mathematics at Kent State University, USA. His research focuses on combinatorics and spectral graph theory. He is currently a Visiting Professor at the Università degli Studi di Napoli, Italy.

Bibliographic Information

  • Book Title: Locating Eigenvalues in Graphs

  • Book Subtitle: Algorithms and Applications

  • Authors: Carlos Hoppen, David P. Jacobs, Vilmar Trevisan

  • Series Title: SpringerBriefs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-031-11698-8

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

  • Softcover ISBN: 978-3-031-11697-1Published: 22 September 2022

  • eBook ISBN: 978-3-031-11698-8Published: 21 September 2022

  • Series ISSN: 2191-8198

  • Series E-ISSN: 2191-8201

  • Edition Number: 1

  • Number of Pages: XII, 136

  • Number of Illustrations: 12 b/w illustrations, 25 illustrations in colour

  • Topics: Linear Algebra, Graph Theory, Algorithms, Discrete Mathematics

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