Extremal Spectral Invariants of Graphs PDF Download
Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download Extremal Spectral Invariants of Graphs PDF full book. Access full book title Extremal Spectral Invariants of Graphs by Robin Joshua Tobin. Download full books in PDF and EPUB format.
Author: Robin Joshua Tobin
Publisher:
ISBN:
Category :
Languages : en
Pages : 92
Get Book Here
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.
Author: Robin Joshua Tobin
Publisher:
ISBN:
Category :
Languages : en
Pages : 92
Get Book Here
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.
Author: Dragan Stevanovic
Publisher: Academic Press
ISBN: 0128020970
Category : Mathematics
Languages : en
Pages : 167
Get Book Here
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
Author: Zoran Stanić
Publisher: Cambridge University Press
ISBN: 1107545978
Category : Mathematics
Languages : en
Pages : 311
Get Book Here
Book Description
This book explores the inequalities for eigenvalues of the six matrices associated with graphs. Includes the main results and selected applications.
Author: Fan R. K. Chung
Publisher: American Mathematical Soc.
ISBN: 0821803158
Category : Mathematics
Languages : en
Pages : 228
Get Book Here
Book Description
This text discusses spectral graph theory.
Author: Sinan Güven Aksoy
Publisher:
ISBN:
Category :
Languages : en
Pages : 109
Get Book Here
Book Description
We apply spectral theory to study random processes involving directed graphs. In the first half of this thesis, we examine random walks on directed graphs, which is rooted in the study of non-reversible Markov chains. We prove bounds on key spectral invariants which play a role in bounding the rate of convergence of the walk and capture isoperimetric properties of the directed graph. We first focus on the principal ratio, which is the ratio of maximum to minimum values of vertices in the stationary distribution. Improving upon previous bounds, we give a sharp upper bound for this ratio over all strongly connected graphs on $n$ vertices. We characterize all graphs achieving the upper bound and give explicit constructions for these extremal graphs. Additionally, we show that under certain conditions, the principal ratio is tightly bounded. We then turn our attention to the first nontrivial Laplacian eigenvalue of a strongly connected directed graph. We give a lower bound for this eigenvalue, extending an analogous result for undirected graphs to the directed case. Our results on the principal ratio imply this lower bound can be factorially small in the number of vertices, and we give a construction having this eigenvalue factorially small. In the second half, we apply spectral tools to study orientations of graphs. We focus on counting orientations yielding strongly connected directed graphs, called strong orientations. Namely, we show that under a mild spectral and minimum degree condition, a possibly irregular, sparse graph $G$ has "many" strong orientations. More precisely, given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if $G$ satisfies a minimum degree condition of $(1+c_1)\log_2{n}$ and has Cheeger constant at least $c_2\frac{\log_2\log_2{n}}{\log_2{n}}$, then the resulting randomly oriented directed graph is strongly connected with high probability. This Cheeger constant bound can be replaced by an analogous spectral condition via the Cheeger inequality. Additionally, we provide an explicit construction to show our minimum degree condition is tight while the Cheeger constant bound is tight up to a $\log_2\log_2{n}$ factor. We conclude by exploring related future work.
Author: Pavel Kurasov
Publisher: Springer Nature
ISBN: 3662678721
Category : Science
Languages : en
Pages : 644
Get Book Here
Book Description
This open access book gives a systematic introduction into the spectral theory of differential operators on metric graphs. Main focus is on the fundamental relations between the spectrum and the geometry of the underlying graph. The book has two central themes: the trace formula and inverse problems. The trace formula is relating the spectrum to the set of periodic orbits and is comparable to the celebrated Selberg and Chazarain-Duistermaat-Guillemin-Melrose trace formulas. Unexpectedly this formula allows one to construct non-trivial crystalline measures and Fourier quasicrystals solving one of the long-standing problems in Fourier analysis. The remarkable story of this mathematical odyssey is presented in the first part of the book. To solve the inverse problem for Schrödinger operators on metric graphs the magnetic boundary control method is introduced. Spectral data depending on the magnetic flux allow one to solve the inverse problem in full generality, this means to reconstruct not only the potential on a given graph, but also the underlying graph itself and the vertex conditions. The book provides an excellent example of recent studies where the interplay between different fields like operator theory, algebraic geometry and number theory, leads to unexpected and sound mathematical results. The book is thought as a graduate course book where every chapter is suitable for a separate lecture and includes problems for home studies. Numerous illuminating examples make it easier to understand new concepts and develop the necessary intuition for further studies.
Author: Dragoš M. Cvetković
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 374
Get Book Here
Book Description
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: Stephan Wagner
Publisher: CRC Press
ISBN: 0429833989
Category : Mathematics
Languages : en
Pages : 252
Get Book Here
Book Description
Introduction to Chemical Graph Theory is a concise introduction to the main topics and techniques in chemical graph theory, specifically the theory of topological indices. These include distance-based, degree-based, and counting-based indices. The book covers some of the most commonly used mathematical approaches in the subject. It is also written with the knowledge that chemical graph theory has many connections to different branches of graph theory (such as extremal graph theory, spectral graph theory). The authors wrote the book in an appealing way that attracts people to chemical graph theory. In doing so, the book is an excellent playground and general reference text on the subject, especially for young mathematicians with a special interest in graph theory. Key Features: A concise introduction to topological indices of graph theory Appealing to specialists and non-specialists alike Provides many techniques from current research About the Authors: Stephan Wagner grew up in Graz (Austria), where he also received his PhD from Graz University of Technology in 2006. Shortly afterwards, he moved to South Africa, where he started his career at Stellenbosch University as a lecturer in January 2007. His research interests lie mostly in combinatorics and related areas, including connections to other scientific fields such as physics, chemistry and computer science. Hua Wang received his PhD from University of South Carolina in 2005. He held a Visiting Research Assistant Professor position at University of Florida before joining Georgia Southern University in 2008. His research interests include combinatorics and graph theory, elementary number theory, and related problems
Author: Andries E. Brouwer
Publisher: Springer Science & Business Media
ISBN: 1461419395
Category : Mathematics
Languages : en
Pages : 254
Get Book Here
Book Description
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: Alex Lubotzky
Publisher: Springer Science & Business Media
ISBN: 3034603320
Category : Mathematics
Languages : en
Pages : 201
Get Book Here
Book Description
In the last ?fteen years two seemingly unrelated problems, one in computer science and the other in measure theory, were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the - plicit construction of expanding graphs («expanders»). These are highly connected sparse graphs whose existence can be easily demonstrated but whose explicit c- struction turns out to be a dif?cult task. Since expanders serve as basic building blocks for various distributed networks, an explicit construction is highly des- able. The other problem is one posed by Ruziewicz about seventy years ago and studied by Banach [Ba]. It asks whether the Lebesgue measure is the only ?nitely additive measure of total measure one, de?ned on the Lebesgue subsets of the n-dimensional sphere and invariant under all rotations. The two problems seem, at ?rst glance, totally unrelated. It is therefore so- what surprising that both problems were solved using similar methods: initially, Kazhdan’s property (T) from representation theory of semi-simple Lie groups was applied in both cases to achieve partial results, and later on, both problems were solved using the (proved) Ramanujan conjecture from the theory of automorphic forms. The fact that representation theory and automorphic forms have anything to do with these problems is a surprise and a hint as well that the two questions are strongly related.
