Download or read book Distance regular antipodal covers of strongly regular graphs written by Aleksandar Jurišić and published by . This book was released on 1990 with total page 73 pages. Available in PDF, EPUB and Kindle. Book excerpt: Antipodal distance regular covers of complete and complete bipartite graphs give rise to projective planes and Hadamard matrices. Inspired by this, we investigate antipodal distance regular covers of strongly regular graphs which are not complete bipartite. We use geometric properties of antipodal distance regular covers to show that infinite families of strongly regular graphs (Steiner graphs and latin square graphs) have no antipodal distance regular covers. This implies that for fixed integer $m$ only finitely many strongly regular graphs with the smallest eigenvalue $-m$ can have distance regular intipodal covers. We search for feasible parameters of strongly regular graphs with small valency (up to 100) and feasible parameters of their antipodal distance regular covers.
Download or read book Antipodal Covers of Strongly Regular Graphs written by Jurǐsi ́c, Aleksandar and published by . This book was released on 1991 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Antipodal Covers of Strongly Regular Graphs written by Aleksandar Jurišić and published by . This book was released on 1991 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Antipodal covers written by Aleksandar Jurišić and published by . This book was released on 1995 with total page 139 pages. Available in PDF, EPUB and Kindle. Book excerpt: A connected graph $G$ is called distance-regular graph if for any two vertices $u$ and $v$ of $G$ at distance $i$, the number $c_i$ (resp. $B-i$) of neighbours of $v$ at distance $i-1$ (resp. $i+1$) from $u$ depends only on $i$ rather than on individual vertices. A distance regular graph $G$ of diameter $d \in \{2m,2m+1\}$ is antipodal ($r$-cover of its folded graph) if and only if $b_i = c_{d-i}$, for $i = 0,...,d$, $i \ne m$ (and $r = 1+b_m/c_{d-m}$). For example, the dodecahedron is a double-cover of the Petersen graph, the cube is the double-cover of the tetrahedron. Most finite objects of sufficient regularity are closed related to certain distance-regular graphs, in particular, antipodal distance-regular graphs give rise to projective planes, Hadamard matrices and other interesting combinatorial objects. Distance-regular graphs serve as an alternative approach to these objects and allow the use of graph eigenvalues, graph representations, association schemes and the theory of orthogonal polynomials. We start with cyclic covers and spreads of generalized quadrangles and find a switching which uses some known infinite families of antipodal distance-regular graphs of diameter three to produce new ones. Then we examine antipodal distance-families of antipodal distance-regular graphs of diameter three to produce new ones. Then we examine antipodal distance-regular graphs of diameter four and five. Together with J. Koolen we use representations of graphs to extend in the case of diameter four the result P. Terwilliger who has shown, using the theory of Krein modules, that in a $Q$-polynomial antipodal diatance-regular graph the neighbourhood of any vertex is a strongly regular graph. New nonexistence conditions for covers are derived from that. This study relates to the above switching and to extended generalized quadrangles. In an imprimitive association scheme there always exists a merging (i.e., a grouping of the relations) which gives a new nontrivial association scheme. We determine when merging in an antipodal distance-regular graph produces a distance-regular graphs with regular near polygons containing a spread. In case of diameter three we get a Brouwer's characterization of certain distance-regular graphs with generalized quadrangles containing a spread. Finally, antipodal covers of strongly regular graphs which are not necessarily distance-regular are studied. In most cases the structure of short cycles provides a tool to determine the existence of an antipodal cover. A relationship between antipodal covers of a graph and its line graph is investigated. Antipodal covers of complete bipartite graphs and their line graphs (lattice graphs) are characterized in terms of weak resolvable transversal designs which are, in the case of maximal covering index, equivalent to affine planes with a parallel class deleted. We conclude by mentioning two results which indicate the importance of antipodal distance-regular graphs. The first one is a joint work with M. Araya and A. Hiraki. Let $G$ be a distance-regular graph of diameter $d$ and valency $k>2$. If $b_t=1$ and $2t \le d$, then $G$ is an antipodal double-cover. Consenquently, if $m>2$ the multiplicity of an eigenvalue of the adjacency matrix of $G$ and if $G$ is not an antipodal double-cover then $d \le 2m-3$. This result is an improvement of Godsil's diameter bound and it is very important for the classification of distance-regular graphs with an eigenvalue of small multiplicity (as opposed to dual classification of distance-regular graphs with small valency). The second result is joint work with C. Godsil. We show that distance-regular graphs that contain maximal independent geodesic paths of short length are antipodal. A new infinite family of feasible parameters of antipodal distance-regular graphs of diameter four is found. As an auxiliary result we use equitable partitions to show that the determinant of a Töplitz matrix can be written as a product of two determinants of approximately half size of the original one.
Download or read book Distance Regular Antipodal Covers of Strongly Regular Graphs written by Chris D. Godsil and published by Faculty of Mathematics, University of Waterloo. This book was released on 1990 with total page 45 pages. Available in PDF, EPUB and Kindle. Book excerpt: We use geometric properties of antipodal distance regular covers to show that infinite families of strongly regular graphs (Steiner graphs and latin square graphs) have no antipodal distance regular covers.
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 Strongly Regular Graphs written by Andries E. Brouwer and published by . This book was released on 2022-01-13 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph on strongly regular graphs is an invaluable reference for anybody working in algebraic combinatorics.
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 Nonexistence of Some Antipodal Distance regular Graphs of Diameter Four written by Aleksandar Jurišić and published by . This book was released on 2000 with total page 8 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Bipartite Distance regular Graphs and Their Primitive Idempotents written by Mark Spruill MacLean and published by . This book was released on 2001 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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.
Download or read book Distance Regular Antipodal Covers of Johnson and Hamming Graphs written by J. T. M. van Bon and published by . This book was released on 1987 with total page 4 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Algebraic Combinatorics written by Eiichi Bannai and published by Walter de Gruyter GmbH & Co KG. This book was released on 2021-02-22 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: This series is devoted to the publication of high-level monographs which cover the whole spectrum of current discrete mathematics and its applications in various fields. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of discrete mathematics. Contributions which are on the borderline of discrete mathematics and related fields and which stimulate further research at the crossroads of these areas are particularly welcome.
Download or read book Finite Generalized Quadrangles written by Stanley E. Payne and published by European Mathematical Society. This book was released on 2009 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Generalized quadrangles (GQ) were formally introduced by J. Tits in 1959 to describe geometric properties of simple groups of Lie type of rank 2. The first edition of Finite Generalized Quadrangles (FGQ) quickly became the standard reference for finite GQ. The second edition is essentially a reprint of the first edition. It is a careful rendering into LaTeX of the original, along with an appendix that brings to the attention of the reader those major new results pertaining to GQ, especially in those areas where the authors of this work have made a contribution. The first edition has been out of print for many years. The new edition makes available again this classical reference in the rapidly increasing field of finite geometries.
Download or read book Distance regular Antipodal Covering Graphs written by Robert Ernest Lee Aldred and published by . This book was released on 1986 with total page 186 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 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.
