Download or read book Antipodal Distance Regular Graphs microform written by Andrew David Hensel and published by National Library of Canada. This book was released on 1988 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Antipodal Distance regular Graphs written by Tilla Schade and published by . This book was released on 1994 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Homotopy and Distance regular Graphs written by Heather Ames Lewis and published by . This book was released on 1997 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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 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 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 Regular T graphs Antipodal Distance Regular Graphs of Diameter 3 and Related Combinatorial Structures written by Daniel Kalmanovich and published by . This book was released on 2011 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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 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 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 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 Imprimitive Distance transitive Graphs written by Monther Rashed Furaidan and published by . This book was released on 2004 with total page 238 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 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 Some Results on Bipartite Distance regular Graphs written by Michael S. Lang and published by . This book was released on 2001 with total page 94 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Bipartite Distance regular Graphs written by Brian Curtin and published by . This book was released on 1996 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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: