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Spectral Radius Of Graphs

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Spectral Radius of Graphs

Spectral Radius of Graphs Book
Author : Dragan Stevanovic
Publisher : Academic Press
Release : 2014-10-13
ISBN : 0128020970
Language : En, Es, Fr & De

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Book Description :

Spectral Radius of Graphs provides a thorough overview of important results on the spectral radius of adjacency matrix of graphs that have appeared in the literature in the preceding ten years, most of them with proofs, and including some previously unpublished results of the author. The primer begins with a brief classical review, in order to provide the reader with a foundation for the subsequent chapters. Topics covered include spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem. From this introduction, the book delves deeper into the properties of the principal eigenvector; a critical subject as many of the results on the spectral radius of graphs rely on the properties of the principal eigenvector for their proofs. A following chapter surveys spectral radius of special graphs, covering multipartite graphs, non-regular graphs, planar graphs, threshold graphs, and others. Finally, the work explores results on the structure of graphs having extreme spectral radius in classes of graphs defined by fixing the value of a particular, integer-valued graph invariant, such as: the diameter, the radius, the domination number, the matching number, the clique number, the independence number, the chromatic number or the sequence of vertex degrees. Throughout, the text includes the valuable addition of proofs to accompany the majority of presented results. This enables the reader to learn tricks of the trade and easily see if some of the techniques apply to a current research problem, without having to spend time on searching for the original articles. The book also contains a handful of open problems on the topic that might provide initiative for the reader's research. Dedicated coverage to one of the most prominent graph eigenvalues Proofs and open problems included for further study Overview of classical topics such as spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem

Spectral Radius and Signless Laplacian Spectral Radius of K connected Graphs

Spectral Radius and Signless Laplacian Spectral Radius of K connected Graphs Book
Author : Peng Huang
Publisher : Unknown
Release : 2016
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

The adjacency matrix of a graph is a (0, 1)-matrix indexed by the vertex set of the graph. And the signless Laplacian matrix of a graph is the sum of its adjacency matrix and its diagonal matrix of vertex degrees. The eigenvalues and the signless Laplacian eigenvalues of a graph are the eigenvalues of the adjacency matrix and the signless Laplacian matrix, respectively. These two matrices of a graph have been studied for several decades since they have been applied to many research field, such as computer science, communication network, information science and so on. In this thesis, we study k-connected graphs and focus on their spectral radius and signless Laplacian spectral radius. Firstly, we determine the graphs with maximum spectral radius among all k-connected graphs of fixed order with given diameter. As we know, when a graph is regular, its spectral radius and signless Laplacian spectral radius can easily be found. We obtain an upper bound on the signless Laplacian spectral radius of k-connected irregular graphs. Finally, we give some other results mainly related to the signless Laplacian matrix.

The Minimal Spectral Radius of Graphs with a Given Diameter

The Minimal Spectral Radius of Graphs with a Given Diameter Book
Author : E.R. van Dam,R.E. Kooij
Publisher : Unknown
Release : 2006
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download The Minimal Spectral Radius of Graphs with a Given Diameter book written by E.R. van Dam,R.E. Kooij, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Laplacian Spectra and Graph Structure

Laplacian Spectra and Graph Structure Book
Author : Kamal Lochan Patra
Publisher : Unknown
Release : 2010
ISBN : 9783843386128
Language : En, Es, Fr & De

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Book Description :

The Laplacian matrix of a graph and its eigenvalues can be used in several areas of mathematical research and have a physical interpretation in various physical and chemical theories. The related matrix, the adjacency matrix of a graph and its eigenvalues were much more investigated in the past than the Laplacian matrix. However in my opinion the Laplacian spectrum is much more natural and more important than the adjacency matrix spectrum. We have used the standard terminology of graph theory, as it is introduced in most text books on the theory of graphs. We have studied the graph structure related to the second smallest and the largest Laplacian eigenvalues -- publisher.

The Minimum Spectral Radius of Graphs with a Given Clique Number

The Minimum Spectral Radius of Graphs with a Given Clique Number Book
Author : Dragan Stevanović,Pierre Hansen,Groupe d'études et de recherche en analyse des décisions (Montréal, Québec)
Publisher : Unknown
Release : 2007
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download The Minimum Spectral Radius of Graphs with a Given Clique Number book written by Dragan Stevanović,Pierre Hansen,Groupe d'études et de recherche en analyse des décisions (Montréal, Québec), available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Inequalities for Graph Eigenvalues

Inequalities for Graph Eigenvalues Book
Author : Zoran Stanić
Publisher : Cambridge University Press
Release : 2015-07-23
ISBN : 1107545978
Language : En, Es, Fr & De

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Book Description :

Explores the inequalities for eigenvalues of the six matrices associated with graphs. Includes the main results and selected applications.

On the Spectral Radius of Graphs with a Given Domination Number

On the Spectral Radius of Graphs with a Given Domination Number Book
Author : Dragan Stevanović,Pierre Hansen,Mustapha Aouchiche,Groupe d'études et de recherche en analyse des décisions (Montréal, Québec)
Publisher : Unknown
Release : 2007
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download On the Spectral Radius of Graphs with a Given Domination Number book written by Dragan Stevanović,Pierre Hansen,Mustapha Aouchiche,Groupe d'études et de recherche en analyse des décisions (Montréal, Québec), available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Graphs with Given Diameter Maximizing the Spectral Radius

Graphs with Given Diameter Maximizing the Spectral Radius Book
Author : E.R. van Dam
Publisher : Unknown
Release : 2006
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Graphs with Given Diameter Maximizing the Spectral Radius book written by E.R. van Dam, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Comparing the Geometric arithmetic Index and the Spectral Radius of Graphs

Comparing the Geometric  arithmetic Index and the Spectral Radius of Graphs Book
Author : Mustapha Aouchiche
Publisher : Unknown
Release : 2016
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Comparing the Geometric arithmetic Index and the Spectral Radius of Graphs book written by Mustapha Aouchiche, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

The Mutually Beneficial Relationship of Graphs and Matrices

The Mutually Beneficial Relationship of Graphs and Matrices Book
Author : Richard A. Brualdi
Publisher : American Mathematical Soc.
Release : 2011-07-06
ISBN : 0821853155
Language : En, Es, Fr & De

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Book Description :

Graphs and matrices enjoy a fascinating and mutually beneficial relationship. This interplay has benefited both graph theory and linear algebra. In one direction, knowledge about one of the graphs that can be associated with a matrix can be used to illuminate matrix properties and to get better information about the matrix. Examples include the use of digraphs to obtain strong results on diagonal dominance and eigenvalue inclusion regions and the use of the Rado-Hall theorem to deduce properties of special classes of matrices. Going the other way, linear algebraic properties of one of the matrices associated with a graph can be used to obtain useful combinatorial information about the graph. The adjacency matrix and the Laplacian matrix are two well-known matrices associated to a graph, and their eigenvalues encode important information about the graph. Another important linear algebraic invariant associated with a graph is the Colin de Verdiere number, which, for instance, characterizes certain topological properties of the graph. This book is not a comprehensive study of graphs and matrices. The particular content of the lectures was chosen for its accessibility, beauty, and current relevance, and for the possibility of enticing the audience to want to learn more.

Extremal Spectral Invariants of Graphs

Extremal Spectral Invariants of Graphs Book
Author : Robin Joshua Tobin
Publisher : Unknown
Release : 2017
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

We address several problems in spectral graph theory, with a common theme of optimizing or computing a spectral graph invariant, such as the spectral radius or spectral gap, over some family of graphs. In particular, we study measures of graph irregularity, we bound the adjacency spectral radius over all outerplanar and planar graphs, and finally we determine the spectral gap of reversal graphs and a family of graphs that generalize the prefix reversal graph. Firstly we study two measures of graph irregularity, the principal ratio and the difference between the spectral radius of the adjacency matrix and the average degree. For the principal ratio, we show that the graphs which maximize this statistic are the kite graphs, which are a clique with a pendant path, when the number of vertices is sufficiently large. This answers a conjecture of Cioabă and Gregory. For the second graph irregularity measure, we show that the connected graphs which maximize it are pineapple graphs, answering a conjecture of Aouchiche et al. Secondly we investigate the maximum spectral radius of the adjacency matrix over all graphs on n vertices within certain well-known graph families. Our main result is showing that the planar graph on n vertices with maximal adjacency spectral radius is the join P 2 + P n-2 , when n is sufficiently large. This was conjectured by Boots and Royle. Additionally, we identify the outerplanar graph with maximal spectral radius, answering a conjecture of Cvetkovic̀ and Rowlinson. Finally, we determine the spectral gap of various Cayley graphs of the symmetric group Sn , which arise in the context of substring reversals. This includes an elementary proof that the prefix reversal (or pancake flipping graph) has spectral gap one, originally proved via representation theory by Cesi. We generalize this by showing that a large family of related graphs all have unit spectral gap.

Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs

Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs Book
Author : Jason J. Molitierno
Publisher : CRC Press
Release : 2012-01-25
ISBN : 1439863377
Language : En, Es, Fr & De

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Book Description :

On the surface, matrix theory and graph theory seem like very different branches of mathematics. However, adjacency, Laplacian, and incidence matrices are commonly used to represent graphs, and many properties of matrices can give us useful information about the structure of graphs. Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs is a compilation of many of the exciting results concerning Laplacian matrices developed since the mid 1970s by well-known mathematicians such as Fallat, Fiedler, Grone, Kirkland, Merris, Mohar, Neumann, Shader, Sunder, and more. The text is complemented by many examples and detailed calculations, and sections followed by exercises to aid the reader in gaining a deeper understanding of the material. Although some exercises are routine, others require a more in-depth analysis of the theorems and ask the reader to prove those that go beyond what was presented in the section. Matrix-graph theory is a fascinating subject that ties together two seemingly unrelated branches of mathematics. Because it makes use of both the combinatorial properties and the numerical properties of a matrix, this area of mathematics is fertile ground for research at the undergraduate, graduate, and professional levels. This book can serve as exploratory literature for the undergraduate student who is just learning how to do mathematical research, a useful "start-up" book for the graduate student beginning research in matrix-graph theory, and a convenient reference for the more experienced researcher.

Graph Theoretic Problems and Their New Applications

Graph Theoretic Problems and Their New Applications Book
Author : Frank Werner
Publisher : MDPI
Release : 2020-05-27
ISBN : 3039287982
Language : En, Es, Fr & De

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Book Description :

Graph theory is an important area of applied mathematics with a broad spectrum of applications in many fields. This book results from aSpecialIssue in the journal Mathematics entitled “Graph-Theoretic Problems and Their New Applications”. It contains 20 articles covering a broad spectrum of graph-theoretic works that were selected from 151 submitted papers after a thorough refereeing process. Among others, it includes a deep survey on mixed graphs and their use for solutions ti scheduling problems. Other subjects include topological indices, domination numbers of graphs, domination games, contraction mappings, and neutrosophic graphs. Several applications of graph theory are discussed, e.g., the use of graph theory in the context of molecular processes.

The Joint Spectral Radius

The Joint Spectral Radius Book
Author : Raphaël Jungers
Publisher : Springer Science & Business Media
Release : 2009-05-19
ISBN : 3540959793
Language : En, Es, Fr & De

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Book Description :

This monograph is based on the Ph.D. Thesis of the author [58]. Its goal is twofold: First, it presents most researchwork that has been done during his Ph.D., or at least the part of the work that is related with the joint spectral radius. This work was concerned with theoretical developments (part I) as well as the study of some applications (part II). As a second goal, it was the author’s feeling that a survey on the state of the art on the joint spectral radius was really missing in the literature, so that the ?rst two chapters of part I present such a survey. The other chapters mainly report personal research, except Chapter 5 which presents animportantapplicationofthejointspectralradius:thecontinuityofwavelet functions. The ?rst part of this monograph is dedicated to theoretical results. The ?rst two chapters present the above mentioned survey on the joint spectral radius. Its minimum-growth counterpart, the joint spectral subradius, is also considered. The next two chapters point out two speci?c theoretical topics, that are important in practical applications: the particular case of nonne- tive matrices, and the Finiteness Property. The second part considers applications involving the joint spectral radius.

Amenability Unimodularity and the Spectral Radius of Random Walks on Infinite Graphs

Amenability  Unimodularity  and the Spectral Radius of Random Walks on Infinite Graphs Book
Author : Paolo M. Soardi,Wolfgang Woess
Publisher : Unknown
Release : 1988
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Amenability Unimodularity and the Spectral Radius of Random Walks on Infinite Graphs book written by Paolo M. Soardi,Wolfgang Woess, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Asymptotic Results on the Spectral Radius and the Diameter of Graphs

Asymptotic Results on the Spectral Radius and the Diameter of Graphs Book
Author : Sebastian M. Cioabǎ
Publisher : Unknown
Release : 2008
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Asymptotic Results on the Spectral Radius and the Diameter of Graphs book written by Sebastian M. Cioabǎ, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Surveys in Combinatorics 2011

Surveys in Combinatorics 2011 Book
Author : Robin Chapman
Publisher : Cambridge University Press
Release : 2011-06-23
ISBN : 1139503685
Language : En, Es, Fr & De

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Book Description :

This volume contains nine survey articles based on the invited lectures given at the 23rd British Combinatorial Conference, held at Exeter in July 2011. This biennial conference is a well-established international event, with speakers from all over the world. By its nature, this volume provides an up-to-date overview of current research activity in several areas of combinatorics, including extremal graph theory, the cyclic sieving phenomenon and transversals in Latin squares. Each article is clearly written and assumes little prior knowledge on the part of the reader. The authors are some of the world's foremost researchers in their fields, and here they summarise existing results and give a unique preview of the most recent developments. The book provides a valuable survey of the present state of knowledge in combinatorics. It will be useful to research workers and advanced graduate students, primarily in mathematics but also in computer science and statistics.

Regular Graphs

Regular Graphs Book
Author : Zoran Stanić
Publisher : Walter de Gruyter GmbH & Co KG
Release : 2017-04-24
ISBN : 311035134X
Language : En, Es, Fr & De

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Book Description :

Written for mathematicians working with the theory of graph spectra, this (primarily theoretical) book presents relevant results considering the spectral properties of regular graphs. The book begins with a short introduction including necessary terminology and notation. The author then proceeds with basic properties, specific subclasses of regular graphs (like distance-regular graphs, strongly regular graphs, various designs or expanders) and determining particular regular graphs. Each chapter contains detailed proofs, discussions, comparisons, examples, exercises and also indicates possible applications. Finally, the author also includes some conjectures and open problems to promote further research. Contents Spectral properties Particular types of regular graph Determinations of regular graphs Expanders Distance matrix of regular graphs

Quantitative Logic and Soft Computing 2016

Quantitative Logic and Soft Computing 2016 Book
Author : Tai-He Fan,Shui-Li Chen,San-Min Wang,Yong-Ming Li
Publisher : Springer
Release : 2016-09-21
ISBN : 3319462067
Language : En, Es, Fr & De

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Book Description :

This book is the proceedings of the Fourth International Conference on Quantitative Logic and Soft Computing (QLSC2016) held 14-17, October, 2016 in Zhejiang Sci-Tech University, Hangzhou, China. It includes 61 papers, of which 5 are plenary talks( 3 abstracts and 2 full length talks). QLSC2016 was the fourth in a series of conferences on Quantitative Logic and Soft Computing. This conference was a major symposium for scientists, engineers and practitioners to present their updated results, ideas, developments and applications in all areas of quantitative logic and soft computing. The book aims to strengthen relations between industry research laboratories and universities in fields such as quantitative logic and soft computing worldwide as follows: (1) Quantitative Logic and Uncertainty Logic; (2) Automata and Quantification of Software; (3) Fuzzy Connectives and Fuzzy Reasoning; (4) Fuzzy Logical Algebras; (5) Artificial Intelligence and Soft Computing; (6) Fuzzy Sets Theory and Applications.