Download or read book On Some Spectral and Combinatorial Properties of Distance regular Graphs and Their Generalizations written by Víctor Diego and published by . This book was released on 2018 with total page 88 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work we present the study we did in Graph Theory. In the firsts chapteres of the tesis we study the pieces of information that can be obtained from a graph: the spectrum of the adjacency matrix, the preintersection numbers, the predistance polynomials and the average number of closed walks. Some of this pieces of information are direct generalizations of the intersection numbers or the predistance polynomials defined in the distance-regular graphs. We prove that the multiple properties that these pieces of information have in distance-regular graphs hold also in their generalizations, and these properties can be applied to any other graph. We also prove that the distinct pieces of information (even if their nature is algebraic or combinatorial) are equivalent. That is, we can obtain each one of the pieces in terms of each other; proving in this way that the properties of the graph derived from each one of the pieces can be also obtained in terms of each one of the other. We dedicate a chapter of the tesis to describe completly the especific procedures with which obtain each piece of inforation in terms of the others. In this tesis we introduce the "distance mean-regular" graphs. These graphs are a generalization of the distance-regular graphs. In this occasion, we demand to the graph combinatorial properties and we generalizate the algebraic properties of the distance-regular graphs. We generalizate the spectrum of a graph to introduce the "pseudo-spectrum" and we generalizate the Bose-Mesner algebra in distinct matrix algebras. The study of these generalizations, as well as the study of the relation between them, give us combinatorial and algebraic properties. In the final part of the tesis we study the vertex-isoperimetric problem in the Johnson Graph J(n,m). We solve completly the problem for some particular cases: J(n,1), J(n,2), J(2m-2,m), as well as their symetrics J(n,n-2) and J(2m+2,m). The solution for these cases are the initial segments of the colexicographic order. This order is also the solution for small cardinals in every graph of this family, as well as for the asymptotic behaviour of the parameters n and m. However, this solution is not the optimal solution for every cardinal in every graph J(n,m). We prove and give an infinity family of counterexamples for which the initial segment of the colexicographic order is not optimal in terms of the vertex-isoperimetric problem.
Download or read book Distance Regular Graphs written by Andries E. Brouwer and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 513 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Regular Graphs written by Zoran Stanić and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-04-24 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Download or read book Combinatorial and Spectral Properties of Graphs and Association Schemes written by Matt McGinnis and published by . This book was released on 2018 with total page 111 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main topics of this dissertation are related to spectral graph theory, a subtopic of algebraic combinatorics. Algebraic combinatorics is the area of mathematics that implements techniques from linear and abstract algebra to solve problems in combinatorics as well as techniques from combinatorics to study algebraic structures. Spectral graph theory focuses on the study of the eigenvalues associated with various matrices for a graph and how the eigenvalues relate to structural properties of the graph. Properties such as connectedness, diameter, independence number, chromatic number and regularity, among others, are all related to the spectrum of a graph. In this dissertation we will study the spectra of various graphs and incorporate well known techniques in spectral graph theory to gain a better understanding of the structure of these graphs. We focus on three topics (Chapters 2, 3 and 4). The variation of these topics reinforces how diverse and useful spectral techniques in graph theory can be. ☐ In Chapter 1 we cover notation and basic definitions used throughout this dissertation. We also introduce some well known and powerful results relating the structural properties of a graph to its spectrum. This includes a discussion of the properties of graphs that can be established from the spectrum as well as which graphs are determined by their spectrum. Finally, we give definitions and basic results for association schemes and distance-regular graphs since the results of this dissertation are related to these structures. ☐ In Chapter 2 we study the smallest eigenvalues for distance-j graphs in both the Hamming and Johnson association schemes. Our results for distance-j Hamming graphs settle a conjecture proposed by Van Dam and Sotirov in [29]. In fact, we reach a stronger conclusion than the one proposed by Van Dam and Sotirov. Our results for x distance-j Johnson graphs settle a conjecture proposed by Karloff in [46]. Again, we are able to obtain a stronger conslusion than what is presented in Karloff's conjecture. ☐ In Chapter 3 we use the technique of Godsil-McKay switching to construct cospectral mates for graphs formed by taking the union of relations in the Johnson association scheme. Our results offer insight into which graphs in this scheme are not determined by their spectrum. Our work also unifies the switching sets previously found for Johnson graphs in [26] and Kneser graphs in [43]. We also present some open problems related to our work, including a switching set that we would like to see generalized in order to obtain a new infinite family of graphs in the Johnson scheme that are not determined by their spectrum. ☐ In Chapter 4 we examine connectivity properties of distance-regular graphs and graphs related to association schemes. In particular, we prove a result on the minimum number of edges that need to be deleted from a distance-regular graph in order to disconnect it into nonsingleton components. We also prove a result on the edge-connectivity of distance-j twisted Grassmann graphs which supports a conjecture proposed by Godsil in [35]. Finally, we end the chapter by presenting open problems dealing with the connectivity of color classes in association schemes.
Download or read book Spectral Generalizations of Line Graphs written by Dragoš Cvetkovic and published by Cambridge University Press. This book was released on 2004-07-22 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction -- Forbidden subgraphs -- Root systems -- Regular graphs -- Star complements -- The Maximal exceptional graphs -- Miscellaneous results.
Download or read book Recent Results in the Theory of Graph Spectra written by D.M. Cvetkovic and published by Elsevier. This book was released on 1988-01-01 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this volume is to review the results in spectral graph theory which have appeared since 1978. The problem of characterizing graphs with least eigenvalue -2 was one of the original problems of spectral graph theory. The techniques used in the investigation of this problem have continued to be useful in other contexts including forbidden subgraph techniques as well as geometric methods involving root systems. In the meantime, the particular problem giving rise to these methods has been solved almost completely. This is indicated in Chapter 1. The study of various combinatorial objects (including distance regular and distance transitive graphs, association schemes, and block designs) have made use of eigenvalue techniques, usually as a method to show the nonexistence of objects with certain parameters. The basic method is to construct a graph which contains the structure of the combinatorial object and then to use the properties of the eigenvalues of the graph. Methods of this type are given in Chapter 2. Several topics have been included in Chapter 3, including the relationships between the spectrum and automorphism group of a graph, the graph isomorphism and the graph reconstruction problem, spectra of random graphs, and the Shannon capacity problem. Some graph polynomials related to the characteristic polynomial are described in Chapter 4. These include the matching, distance, and permanental polynomials. Applications of the theory of graph spectra to Chemistry and other branches of science are described from a mathematical viewpoint in Chapter 5. The last chapter is devoted to the extension of the theory of graph spectra to infinite graphs.
Download or read book Spectral Generalizations of Line Graphs written by Dragoš M. Cvetković and published by . This book was released on 2014-05-14 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Line graphs have the property that their least eigenvalue is greater than, or equal to, -2, a property shared by generalized line graphs and a finite number of so-called exceptional graphs. This book deals with all these families of graphs in the context of their spectral properties. Technical descriptions of these graphs are included in the appendices, while the bibliography provides over 250 references. It will be an important resource for all researchers with an interest in algebraic graph theory.
Download or read book Distance regular Graphs and Generalizations written by Paul M. Terwilliger and published by . This book was released on 1982 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Journal of Combinatorial Theory written by and published by . This book was released on 1996 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Some Problems in the Theory of Distance regular Graphs written by Benjamin V. C. Collins and published by . This book was released on 1996 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Spectral Characterizations of Some Distance regular Graphs written by Edwin R. van Dam and published by . This book was released on 2000 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Spectra of Graphs written by Dragoš M. Cvetković and published by . This book was released on 1980 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Spectral Generalizations of Line Graphs written by Dragoš M. Cvetković and published by . This book was released on 2004 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: An important resource for all researchers with an interest in algebraic graph theory.
Download or read book Regular Graphs written by Zoran Stanić and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-04-24 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Download or read book The Distance regular Graphs with an Eigenvalue of Multiplicity Four written by University of Waterloo. Department of Combinatorics and Optimization and published by . This book was released on 1989 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Bipartite Distance Regular Graphs of Diameter Four written by Junbo Huang and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Using a method by Godsil and Roy, bipartite distance-regular graphs of diameter four can be used to construct $\{0,\alpha\}$-sets, a generalization of the widely applied equiangular sets and mutually unbiased bases. In this thesis, we study the properties of these graphs. There are three main themes of the thesis. The first is the connection between bipartite distance-regular graphs of diameter four and their halved graphs, which are necessarily strongly regular. We derive formulae relating the parameters of a graph of diameter four to those of its halved graphs, and use these formulae to derive a necessary condition for the point graph of a partial geometry to be a halved graph. Using this necessary condition, we prove that several important families of strongly regular graphs cannot be halved graphs. The second theme is the algebraic properties of the graphs. We study Krein parameters as the first part of this theme. We show that bipartite-distance regular graphs of diameter four have one "special" Krein parameter, denoted by $\krein$. We show that the antipodal bipartite distance-regular graphs of diameter four with $\krein=0$ are precisely the Hadamard graphs. In general, we show that a bipartite distance-regular graph of diameter four satisfies $\krein=0$ if and only if it satisfies the so-called $Q$-polynomial property. In relation to halved graphs, we derive simple formulae for computing the Krein parameters of a halved graph in terms of those of the bipartite graph. As the second part of the algebraic theme, we study Terwilliger algebras. We describe all the irreducible modules of the complex space under the Terwilliger algebra of a bipartite distance-regular graph of diameter four, and prove that no irreducible module can contain two linearly independent eigenvectors of the graph with the same eigenvalue. Finally, we study constructions and bounds of $\{0,\alpha\}$-sets as the third theme. We present some distance-regular graphs that provide new constructions of $\{0,\alpha\}$-sets. We prove bounds for the sizes of $\{0,\alpha\}$-sets of flat vectors, and characterize all the distance-regular graphs that yield $\{0,\alpha\}$-sets meeting the bounds at equality. We also study bipartite covers of linear Cayley graphs, and present a geometric condition and a coding theoretic condition for such a cover to produce $\{0,\alpha\}$-sets. Using simple operations on graphs, we show how new $\{0,\alpha\}$-sets can be constructed from old ones.
Download or read book Spectra of Graphs written by Andries E. Brouwer and published by Springer Science & Business Media. This book was released on 2011-12-17 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
