On Strongly Regular Graphs
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Strongly Regular Graphs
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Author : Andries E. Brouwer
language : en
Publisher:
Release Date : 2022-01-13
Strongly Regular Graphs written by Andries E. Brouwer and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-01-13 with Language Arts & Disciplines categories.
This monograph on strongly regular graphs is an invaluable reference for anybody working in algebraic combinatorics.
On Strongly Regular Graphs
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Author : Majid Behbahani
language : en
Publisher:
Release Date : 2009
On Strongly Regular Graphs written by Majid Behbahani and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with categories.
Strongly regular graphs are regular graphs with the additional property that the number of common neighbours for two vertices depends only on whether the vertices are adjacent or non-adjacent. From an algebraic point of view, a graph is strongly regular if its adjacency matrix has exactly three eigenvalues. Strongly regular graphs have very interesting algebraic properties due to their strong regularity conditions. Many strongly regular graphs are known to have large and interesting automorphism groups [23]. In [23] it is also conjectured that almost all strongly regular graphs are asymmetric. Peter Cameron in [7] mentions that "Strongly regular graphs lie on the cusp between highly structured and unstructured." Although strongly regular graphs have been studied extensively since they were introduced, there is very little known about the automorphism group of an arbitrary strongly regular graph based on its parameters. In this thesis, we have developed theory for studying the automorphisms of strongly regular graphs. Our study is both mathematical and computational. On the computational side, we introduce the notion of orbit matrices. Using these matrices, we were able to show that some strongly regular graphs do not admit an automorphism of a certain order. Given the size of the automorphism, we can generate all of the orbit matrices, using a computer program. Another computer program is implemented that generates all the strongly regular graphs from that orbit matrix. From a mathematical point of view, we have found an upper bound on the number of fixed points of the automorphisms of a strongly regular graph. This upper bound is a new upper bound and is obtained by algebraic techniques.
Distance Regular Graphs
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Author : Andries E. Brouwer
language : en
Publisher: Springer
Release Date : 2011-12-06
Distance Regular Graphs written by Andries E. Brouwer and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-12-06 with Mathematics categories.
Ever since the discovery of the five platonic solids in ancient times, the study of symmetry and regularity has been one of the most fascinating aspects of mathematics. Quite often the arithmetical regularity properties of an object imply its uniqueness and the existence of many symmetries. This interplay between regularity and symmetry properties of graphs is the theme of this book. Starting from very elementary regularity properties, the concept of a distance-regular graph arises naturally as a common setting for regular graphs which are extremal in one sense or another. Several other important regular combinatorial structures are then shown to be equivalent to special families of distance-regular graphs. Other subjects of more general interest, such as regularity and extremal properties in graphs, association schemes, representations of graphs in euclidean space, groups and geometries of Lie type, groups acting on graphs, and codes are covered independently. Many new results and proofs and more than 750 references increase the encyclopaedic value of this book.
Spectra Of Graphs
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Author : Andries E. Brouwer
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-12-17
Spectra Of Graphs written by Andries E. Brouwer and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-12-17 with Mathematics categories.
This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.
Regular Graphs
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Author : Zoran Stanić
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2017-04-24
Regular Graphs written by Zoran Stanić and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-04-24 with Mathematics categories.
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
Geometry And Combinatorics
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Author : J. J. Seidel
language : en
Publisher: Academic Press
Release Date : 2014-05-10
Geometry And Combinatorics written by J. J. Seidel and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-10 with Mathematics categories.
Geometry and Combinatorics: Selected Works of J. J. Seidel brings together some of the works of J. J. Seidel in geometry and combinatorics. Seidel's selected papers are divided into four areas: graphs and designs; lines with few angles; matrices and forms; and non-Euclidean geometry. A list of all of Seidel's publications is included. Comprised of 29 chapters, this book begins with a discussion on equilateral point sets in elliptic geometry, followed by an analysis of strongly regular graphs of L2-type and of triangular type. The reader is then introduced to strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3; graphs related to exceptional root systems; and equiangular lines. Subsequent chapters deal with the regular two-graph on 276 vertices; the congruence order of the elliptic plane; equi-isoclinic subspaces of Euclidean spaces; and Wielandt's visibility theorem. This monograph will be of interest to students and practitioners in the field of mathematics.
Graphs And Matrices
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Author : Ravindra B. Bapat
language : en
Publisher: Springer
Release Date : 2014-09-19
Graphs And Matrices written by Ravindra B. Bapat and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-19 with Mathematics categories.
This new edition illustrates the power of linear algebra in the study of graphs. The emphasis on matrix techniques is greater than in other texts on algebraic graph theory. Important matrices associated with graphs (for example, incidence, adjacency and Laplacian matrices) are treated in detail. Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph. Such an extensive coverage of the subject area provides a welcome prompt for further exploration. The inclusion of exercises enables practical learning throughout the book. In the new edition, a new chapter is added on the line graph of a tree, while some results in Chapter 6 on Perron-Frobenius theory are reorganized. Whilst this book will be invaluable to students and researchers in graph theory and combinatorial matrix theory, it will also benefit readers in the sciences and engineering.
Algebraic Graph Theory
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Author : Chris Godsil
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-01
Algebraic Graph Theory written by Chris Godsil and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-01 with Mathematics categories.
This book presents and illustrates the main tools and ideas of algebraic graph theory, with a primary emphasis on current rather than classical topics. It is designed to offer self-contained treatment of the topic, with strong emphasis on concrete examples.
Topics In Algebraic Graph Theory
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Author : Lowell W. Beineke
language : en
Publisher: Cambridge University Press
Release Date : 2004-10-04
Topics In Algebraic Graph Theory written by Lowell W. Beineke and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-10-04 with Mathematics categories.
There is no other book with such a wide scope of both areas of algebraic graph theory.