Laplacian Eigenvectors of Graphs

Laplacian Eigenvectors of Graphs
Author: Türker Biyikoglu
Publisher: Springer
Total Pages: 121
Release: 2007-07-07
Genre: Mathematics
ISBN: 3540735100

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This fascinating volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, and graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology. Eigenvectors of graph Laplacians may seem a surprising topic for a book, but the authors show that there are subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs.

Laplacian Eigenvectors of Graphs

Laplacian Eigenvectors of Graphs
Author: Türker Biyikoglu
Publisher: Springer
Total Pages: 120
Release: 2007-07-26
Genre: Mathematics
ISBN: 9783540735090

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This fascinating volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, and graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology. Eigenvectors of graph Laplacians may seem a surprising topic for a book, but the authors show that there are subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs.

Spectral Radius of Graphs

Spectral Radius of Graphs
Author: Dragan Stevanovic
Publisher: Academic Press
Total Pages: 167
Release: 2014-10-13
Genre: Mathematics
ISBN: 0128020970

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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have appeared in the literature in the preceding ten years, most of them with proofs, and including some previously unpublished results of the author.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

Locating Eigenvalues in Graphs

Locating Eigenvalues in Graphs
Author: Carlos Hoppen
Publisher: Springer Nature
Total Pages: 142
Release: 2022-09-21
Genre: Mathematics
ISBN: 3031116984

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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.

Spectra of Graphs

Spectra of Graphs
Author: Dragoš M. Cvetković
Publisher:
Total Pages: 374
Release: 1980
Genre: Mathematics
ISBN:

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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.

Learning Representation and Control in Markov Decision Processes

Learning Representation and Control in Markov Decision Processes
Author: Sridhar Mahadevan
Publisher: Now Publishers Inc
Total Pages: 185
Release: 2009
Genre: Computers
ISBN: 1601982380

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Provides a comprehensive survey of techniques to automatically construct basis functions or features for value function approximation in Markov decision processes and reinforcement learning.

Graph Symmetry

Graph Symmetry
Author: Gena Hahn
Publisher: Springer Science & Business Media
Total Pages: 434
Release: 2013-03-14
Genre: Mathematics
ISBN: 9401589372

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The last decade has seen two parallel developments, one in computer science, the other in mathematics, both dealing with the same kind of combinatorial structures: networks with strong symmetry properties or, in graph-theoretical language, vertex-transitive graphs, in particular their prototypical examples, Cayley graphs. In the design of large interconnection networks it was realised that many of the most fre quently used models for such networks are Cayley graphs of various well-known groups. This has spawned a considerable amount of activity in the study of the combinatorial properties of such graphs. A number of symposia and congresses (such as the bi-annual IWIN, starting in 1991) bear witness to the interest of the computer science community in this subject. On the mathematical side, and independently of any interest in applications, progress in group theory has made it possible to make a realistic attempt at a complete description of vertex-transitive graphs. The classification of the finite simple groups has played an important role in this respect.

Lx = B - Laplacian Solvers and Their Algorithmic Applications

Lx = B - Laplacian Solvers and Their Algorithmic Applications
Author: Nisheeth K Vishnoi
Publisher:
Total Pages: 168
Release: 2013-03-01
Genre:
ISBN: 9781601986566

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Illustrates the emerging paradigm of employing Laplacian solvers to design novel fast algorithms for graph problems through a small but carefully chosen set of examples. This monograph can be used as the text for a graduate-level course, or act as a supplement to a course on spectral graph theory or algorithms.

Graph Embeddings and Laplacian Eigenvalues

Graph Embeddings and Laplacian Eigenvalues
Author: Stephen Guattery
Publisher:
Total Pages: 26
Release: 1998
Genre: Embeddings (Mathematics)
ISBN:

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Abstract: "Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. For an n x n Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian. An embedding can be represented by a matrix [gamma]; the best possible bound based on this embedding is n/[lambda][subscript max]([gamma superscript T gamma]). However, the best bounds produced by embedding techniques are not tight; they can be off by a factor proportional to log2n for some Laplacians. We show that this gap is a result of the representation of the embedding: by including edge directions in the embedding matrix representation [gamma], it is possible to find an embedding such that [gamma superscript T gamma] has eigenvalues that can be put into a one-to-one correspondence with the eigenvalues of the Laplacian. Specifically, if [lambda] is a nonzero eigenvalue of either matrix, then n/[lambda] is an eigenvalue of the other. Simple transformations map the corresponding eigenvectors to each other. The embedding that produces these correspondences has a simple description in electrical terms if the underlying graph of the Laplaciain [sic] is viewed as a resistive circuit. We also show that a similar technique works for star embeddings when the Laplacian has a zero Dirichlet boundary condition, though the related eigenvalues in this case are reciprocals of each other. In the Dirichlet boundary case, the embedding matrix [gamma] can be used to construct the inverse of the Laplacian. Finally, we connect our results with previous techniques for producing bounds, and provide an illustrative example."

Distribution of Laplacian Eigenvalues of Graphs

Distribution of Laplacian Eigenvalues of Graphs
Author: Bilal Ahmad Rather
Publisher: A.K. Publications
Total Pages: 0
Release: 2022-12-22
Genre: Mathematics
ISBN: 9783258974040

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Spectral graph theory (Algebraic graph theory) is the study of spectral properties of matrices associated to graphs. The spectral properties include the study of characteristic polynomial, eigenvalues and eigenvectors of matrices associated to graphs. This also includes the graphs associated to algebraic structures like groups, rings and vector spaces. The major source of research in spectral graph theory has been the study of relationship between the structural and spectral properties of graphs. Another source has research in mathematical chemistry (theoretical/quantum chemistry). One of the major problems in spectral graph theory lies in finding the spectrum of matrices associated to graphs completely or in terms of spectrum of simpler matrices associated with the structure of the graph. Another problem which is worth to mention is to characterise the extremal graphs among all the graphs or among a special class of graphs with respect to a given graph, like spectral radius, the second largest eigenvalue, the smallest eigenvalue, the second smallest eigenvalue, the graph energy and multiplicities of the eigenvalues that can be associated with the graph matrix. The main aim is to discuss the principal properties and structure of a graph from its eigenvalues. It has been observed that the eigenvalues of graphs are closely related to all graph parameters, linking one property to another. Spectral graph theory has a wide range of applications to other areas of mathematical science and to other areas of sciences which include Computer Science, Physics, Chemistry, Biology, Statistics, Engineering etc. The study of graph eigen- values has rich connections with many other areas of mathematics. An important development is the interaction between spectral graph theory and differential geometry. There is an interesting connection between spectral Riemannian geometry and spectral graph theory. Graph operations help in partitioning of the embedding space, maximising inter-cluster affinity and minimising inter-cluster proximity. Spectral graph theory plays a major role in deforming the embedding spaces in geometry. Graph spectra helps us in making conclusions that we cannot recognize the shapes of solids by their sounds. Algebraic spectral methods are also useful in studying the groups and the rings in a new light. This new developing field investigates the spectrum of graphs associated with the algebraic structures like groups and rings. The main motive to study these algebraic structures graphically using spectral analysis is to explore several properties of interest.