Graphs With Eigenvalues Of High Multiplicity
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| Author | : Casey Boyett |
| Publisher | : |
| Total Pages | : 141 |
| Release | : 2015 |
| Genre | : Electronic dissertations |
| ISBN | : |
Given a graph G we can form a matrix A[subscript G] indexed by the vertices of G and which encodes the edges of G. A[subscript G] is called the adjacency matrix of G. From the adjacency matrix we may find the eigenvalues. We would now like to know what information we may garner from the eigenvalues. It turns out quite a bit may be determined from the eigenvalues, collectively called the spectrum. One big question is to ask whether or not a graph can be uniquely determined by its spectrum. Much research has been done in this area, and it is conjectured that almost all graphs may in fact be determined by their spectra. This is however a difficult task. In this dissertation we look at a subset of all graphs, namely those with either -1 or 0 in their spectrum with a given multiplicity. We first show that any such graph must either be primitive in a sense, or that it is obtained from a primitive graph by an elementary operation of blowing up or splitting vertices. We then show that the set of primitive graphs is finite, for a fixed multiplicity. Lastly, we analyze graphs with -1 or 0 in their spectra with multiplicities up to 4, and show many which are uniquely determined by their spectra.
| Author | : Charles R. Johnson |
| Publisher | : Cambridge University Press |
| Total Pages | : 315 |
| Release | : 2018-02-12 |
| Genre | : Mathematics |
| ISBN | : 1108547036 |
The arrangement of nonzero entries of a matrix, described by the graph of the matrix, limits the possible geometric multiplicities of the eigenvalues, which are far more limited by this information than algebraic multiplicities or the numerical values of the eigenvalues. This book gives a unified development of how the graph of a symmetric matrix influences the possible multiplicities of its eigenvalues. While the theory is richest in cases where the graph is a tree, work on eigenvalues, multiplicities and graphs has provided the opportunity to identify which ideas have analogs for non-trees, and those for which trees are essential. It gathers and organizes the fundamental ideas to allow students and researchers to easily access and investigate the many interesting questions in the subject.
| Author | : Charles R. Johnson |
| Publisher | : Cambridge University Press |
| Total Pages | : 315 |
| Release | : 2018-02-12 |
| Genre | : Mathematics |
| ISBN | : 110709545X |
This book investigates the influence of the graph of a symmetric matrix on the multiplicities of its eigenvalues.
| Author | : Piet van Mieghem |
| Publisher | : Cambridge University Press |
| Total Pages | : 363 |
| Release | : 2010-12-02 |
| Genre | : Technology & Engineering |
| ISBN | : 1139492276 |
Analyzing the behavior of complex networks is an important element in the design of new man-made structures such as communication systems and biologically engineered molecules. Because any complex network can be represented by a graph, and therefore in turn by a matrix, graph theory has become a powerful tool in the investigation of network performance. This self-contained 2010 book provides a concise introduction to the theory of graph spectra and its applications to the study of complex networks. Covering a range of types of graphs and topics important to the analysis of complex systems, this guide provides the mathematical foundation needed to understand and apply spectral insight to real-world systems. In particular, the general properties of both the adjacency and Laplacian spectrum of graphs are derived and applied to complex networks. An ideal resource for researchers and students in communications networking as well as in physics and mathematics.
| Author | : Dragoš M. Cvetković |
| Publisher | : |
| Total Pages | : 374 |
| Release | : 1980 |
| Genre | : Mathematics |
| ISBN | : |
The theory of graph spectra can, in a way, be considered as an attempt to utilize linear algebra including, in particular, the well-developed theory of matrices for the purposes of graph theory and its applications. to the theory of matrices; on the contrary, it has its own characteristic features and specific ways of reasoning fully justifying it to be treated as a theory in its own right.
| Author | : Richard A. Brualdi |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 266 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461383544 |
This IMA Volume in Mathematics and its Applications COMBINATORIAL AND GRAPH-THEORETICAL PROBLEMS IN LINEAR ALGEBRA is based on the proceedings of a workshop that was an integral part of the 1991-92 IMA program on "Applied Linear Algebra." We are grateful to Richard Brualdi, George Cybenko, Alan George, Gene Golub, Mitchell Luskin, and Paul Van Dooren for planning and implementing the year-long program. We especially thank Richard Brualdi, Shmuel Friedland, and Victor Klee for organizing this workshop and editing the proceedings. The financial support of the National Science Foundation made the workshop possible. A vner Friedman Willard Miller, Jr. PREFACE The 1991-1992 program of the Institute for Mathematics and its Applications (IMA) was Applied Linear Algebra. As part of this program, a workshop on Com binatorial and Graph-theoretical Problems in Linear Algebra was held on November 11-15, 1991. The purpose of the workshop was to bring together in an informal setting the diverse group of people who work on problems in linear algebra and matrix theory in which combinatorial or graph~theoretic analysis is a major com ponent. Many of the participants of the workshop enjoyed the hospitality of the IMA for the entire fall quarter, in which the emphasis was discrete matrix analysis.
| Author | : Martin, W. J. (William Joseph) |
| Publisher | : |
| Total Pages | : 24 |
| Release | : 1992 |
| Genre | : Eigenvalues |
| ISBN | : |
| Author | : University of Waterloo. Department of Combinatorics and Optimization |
| Publisher | : |
| Total Pages | : 24 |
| Release | : 1989 |
| Genre | : Eigenvalues |
| ISBN | : |
| Author | : Andries E. Brouwer |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 254 |
| Release | : 2011-12-17 |
| Genre | : Mathematics |
| ISBN | : 1461419395 |
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.
| Author | : Dragoš Cvetković |
| Publisher | : Cambridge University Press |
| Total Pages | : 0 |
| Release | : 2009-10-15 |
| Genre | : Mathematics |
| ISBN | : 9780521134088 |
This introductory text explores the theory of graph spectra: a topic with applications across a wide range of subjects, including computer science, quantum chemistry and electrical engineering. The spectra examined here are those of the adjacency matrix, the Seidel matrix, the Laplacian, the normalized Laplacian and the signless Laplacian of a finite simple graph. The underlying theme of the book is the relation between the eigenvalues and structure of a graph. Designed as an introductory text for graduate students, or anyone using the theory of graph spectra, this self-contained treatment assumes only a little knowledge of graph theory and linear algebra. The authors include many new developments in the field which arise as a result of rapidly expanding interest in the area. Exercises, spectral data and proofs of required results are also provided. The end-of-chapter notes serve as a practical guide to the extensive bibliography of over 500 items.
