Locating Eigenvalues in Graphs

Algorithms and Applications

Carlos Hoppen, David P. Jacobs, Vilmar Trevisan

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English

Springer International Publishing AG
22 September 2022

Discrete mathematics; Algebra; Combinatorics & graph theory; Algorithms & data structures

Series: SpringerBriefs in Mathematics

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.

By: Carlos Hoppen, David P. Jacobs, Vilmar Trevisan
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 1st ed. 2022
Dimensions: Height: 235mm, Width: 155mm,
Weight: 238g
ISBN: 9783031116971
ISBN 10: 3031116976
Series: SpringerBriefs in Mathematics
Pages: 136
Publication Date: 22 September 2022
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

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.

Reviews for Locating Eigenvalues in Graphs: Algorithms and Applications

“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)