Two Adventures in Spectral Graph Theory PDF Download
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Author: Danielle Rogers
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Author: Danielle Rogers
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Author: W. David Joyner
Publisher: Birkhäuser
ISBN: 3319683837
Category : Mathematics
Languages : en
Pages : 344
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Book Description
This textbook acts as a pathway to higher mathematics by seeking and illuminating the connections between graph theory and diverse fields of mathematics, such as calculus on manifolds, group theory, algebraic curves, Fourier analysis, cryptography and other areas of combinatorics. An overview of graph theory definitions and polynomial invariants for graphs prepares the reader for the subsequent dive into the applications of graph theory. To pique the reader’s interest in areas of possible exploration, recent results in mathematics appear throughout the book, accompanied with examples of related graphs, how they arise, and what their valuable uses are. The consequences of graph theory covered by the authors are complicated and far-reaching, so topics are always exhibited in a user-friendly manner with copious graphs, exercises, and Sage code for the computation of equations. Samples of the book’s source code can be found at github.com/springer-math/adventures-in-graph-theory. The text is geared towards advanced undergraduate and graduate students and is particularly useful for those trying to decide what type of problem to tackle for their dissertation. This book can also serve as a reference for anyone interested in exploring how they can apply graph theory to other parts of mathematics.
Author: William L. William L. Hamilton
Publisher: Springer Nature
ISBN: 3031015886
Category : Computers
Languages : en
Pages : 141
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Book Description
Graph-structured data is ubiquitous throughout the natural and social sciences, from telecommunication networks to quantum chemistry. Building relational inductive biases into deep learning architectures is crucial for creating systems that can learn, reason, and generalize from this kind of data. Recent years have seen a surge in research on graph representation learning, including techniques for deep graph embeddings, generalizations of convolutional neural networks to graph-structured data, and neural message-passing approaches inspired by belief propagation. These advances in graph representation learning have led to new state-of-the-art results in numerous domains, including chemical synthesis, 3D vision, recommender systems, question answering, and social network analysis. This book provides a synthesis and overview of graph representation learning. It begins with a discussion of the goals of graph representation learning as well as key methodological foundations in graph theory and network analysis. Following this, the book introduces and reviews methods for learning node embeddings, including random-walk-based methods and applications to knowledge graphs. It then provides a technical synthesis and introduction to the highly successful graph neural network (GNN) formalism, which has become a dominant and fast-growing paradigm for deep learning with graph data. The book concludes with a synthesis of recent advancements in deep generative models for graphs—a nascent but quickly growing subset of graph representation learning.
Author: Yesim Demiroğlu Karabulut
Publisher:
ISBN:
Category :
Languages : en
Pages : 96
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"In the first half of this thesis, we obtain sharp results for Waring's problem over general finite rings, by using a combination of Artin-Wedderburn theory and Hensel's lemma and building on new proofs of analogous results over finite fields that are achieved using spectral graph theory. We also prove an analogue of Sárközy's theorem for finite fields. In the second half of the thesis, we investigate the unit-graphs and the special unit-digraphs on matrix rings and we show that every n x n nonzero matrix over Fq can be written as a sum of two SLn-matrices when n > 1. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties. We prove that if X is a subset of Mat2(Fq) with size [equation would not render] then X contains at least two distinct matrices whose difference has determinant for any [equation would not render]. Using this result we also prove a sum-product type result: if A,B,C;D[subset]Fq satisfy [equation would not render] as q[rightarrow][infinity], then (A-B)(C-D) equals all of F*q. In particular, if A is a subset of Fq with cardinality |A| > 3/2 q 3/4, then the subset (A - A)(A - A) equals all of Fq. We also recover some classical results, e.g. every element in any finite ring of odd order can be written as the sum of two units, and we also derive some character sum identities."--Page vii.
Author: Xiaogang Liu
Publisher:
ISBN:
Category : Cayley graphs
Languages : en
Pages : 148
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Author: Fan R. K. Chung
Publisher:
ISBN: 9780821803158
Category :
Languages : en
Pages : 212
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Book Description
Author: Olaf Post
Publisher: Springer Science & Business Media
ISBN: 3642238394
Category : Mathematics
Languages : en
Pages : 444
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Book Description
Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.
Author: Nair Abreu
Publisher:
ISBN:
Category :
Languages : en
Pages : 177
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Author: Lars Elden
Publisher: SIAM
ISBN: 1611975867
Category : Mathematics
Languages : en
Pages : 229
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Book Description
This thoroughly revised second edition provides an updated treatment of numerical linear algebra techniques for solving problems in data mining and pattern recognition. Adopting an application-oriented approach, the author introduces matrix theory and decompositions, describes how modern matrix methods can be applied in real life scenarios, and provides a set of tools that students can modify for a particular application. Building on material from the first edition, the author discusses basic graph concepts and their matrix counterparts. He introduces the graph Laplacian and properties of its eigenvectors needed in spectral partitioning and describes spectral graph partitioning applied to social networks and text classification. Examples are included to help readers visualize the results. This new edition also presents matrix-based methods that underlie many of the algorithms used for big data. The book provides a solid foundation to further explore related topics and presents applications such as classification of handwritten digits, text mining, text summarization, PageRank computations related to the Google search engine, and facial recognition. Exercises and computer assignments are available on a Web page that supplements the book. This book is primarily for undergraduate students who have previously taken an introductory scientific computing/numerical analysis course and graduate students in data mining and pattern recognition areas who need an introduction to linear algebra techniques.
Author: Robin Joshua Tobin
Publisher:
ISBN:
Category :
Languages : en
Pages : 92
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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.
