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On the geometry of certain strongly regular graphs.

Jalal Naoum Kassab

On the geometry of certain strongly regular graphs.

by Jalal Naoum Kassab

  • 117 Want to read
  • 37 Currently reading

Published by University of Birmingham in Birmingham .
Written in English


Edition Notes

Thesis (Ph.D.) - University of Birmingham, Dept of Pure Mathematics.

ID Numbers
Open LibraryOL13796351M

n are not strongly regular. Strongly regular graphs, tracing back to Bose [6], have been extensively studied from various points of view, for example see [9,10,15] as well as the large list [7]. In this section we are interested in the QE constant of a strongly regular graph. Note that for a connected strongly regular graph G we have n ≥ 4, 2. In mathematics, graphs are useful in geometry and certain parts of topology such as knot theory. Algebraic graph theory has close links with group theory. Algebraic graph theory has been applied to many areas including dynamic systems and complexity. Other topics. A graph structure can be extended by assigning a weight to each edge of the graph.

Section Planar Graphs Investigate! When a connected graph can be drawn without any edges crossing, it is called a planar graph is drawn in this way, it divides the plane into regions called faces.. Draw, if possible, two different planar graphs with the . Geometry (from the Ancient Greek: γεωμετρία; geo-"earth", -metron "measurement") is, with arithmetic, one of the oldest branches of is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.. Until the 19th century, geometry was exclusively.

A (semi)partial geometry is itself a generalisation of the concept of a generalized quadrangle. F. De Clerck, J. A. Thas and collaborators have published several important results on new constructions and characterisations of these geometries. The links with the theory of strongly regular graphs and design theory are very interesting. This is a Wikipedia Book, a collection of articles which can be downloaded electronically or ordered in dia Books are maintained by the Wikipedia community, particularly WikiProject dia Books can also be tagged by the banners of any relevant Wikiprojects (with |class=book). Book This redirect does not require a rating on the project's quality scale.


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On the geometry of certain strongly regular graphs by Jalal Naoum Kassab Download PDF EPUB FB2

Comprised of 29 chapters, this book begins with a discussion on On the geometry of certain strongly regular graphs. book point sets in elliptic geometry, followed by an analysis of strongly regular graphs of L2-type and of triangular type.

The reader is then introduced to strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3; graphs related to exceptional root systems; and equiangular lines.

This chapter itself can be regarded as a self-contained introduction to strongly regular and distance-regular graphs. This book is essentially self-contained, only assuming the knowledge of basic notions from (linear) algebra and projective and affine geometry.

Almost all theorems are accompanied with proofs and a list of exercises with full solutions is given at the end of the book. The point graph of the geometry. Each precisely 4-isoregular graph is a pseudogeometric graph.

An amply regular graph with certain We prove that a strongly regular graph Γ with parameters. The graphs C a y (F p 2 ℓ s, D i), 1 ⩽ i ⩽ p ℓ + 1, are all negative Latin square type or all Latin square type strongly regular graphs according as s is even or odd.

Then, C a y (F p 2 ℓ (s + 1) × F p 2 ℓ s, D) is a negative Latin square type strongly regular graph, where D = ⋃ i = 1 p ℓ Cited by: 2. The Paley graphs and other strongly regular (and similar) graphs have been used as models of \pseudo-random graphs" (see Thomason [28]).

Recently, Fon-Der-Flaass [14] has observed that an old construction of Wallis [29] gives rise to more than exponentially many strongly regular graphs with various parameter sets to be discussed below.

An approach to the enumeration of feasible parameters for strongly regular graphs is described, based on the pair of structural parameters (a,c) and the positive eigenvalue e.

Book description. The projective and polar geometries that arise from a vector space over a finite field are particularly useful in the construction of combinatorial objects, such as latin squares, designs.

Note that any rank 3 graph is a strongly regular graph. The converse is not always true. The complementary graph of a strongly regular graph with parameters (n,k,λ,µ) is a strongly regular graph with parameters (n,n− k − 1,n−2k + µ− 2,n−2k +λ).

A code CG of a graph G is the code of its (0,1)-adjacency matrix. The dimension of CG. A new partial geometry with parameters (s,t,α) = (7,8,4), J. Geometry 16 () K. Coolsaet, некоторых классов сильно регулярных графов = The construction and properties of certain classes of strongly regular graphs (Russian), Uspehi Mat.

Tomorrow's answer's today. Find correct step-by-step solutions for ALL your homework for FREE. To keep things short I skip some definitions on strongly-regular graphs. All necessary information is contained in the paper (see section on strongly-regular graphs).

Bondarenko uses a representation of strongly regular graphs to construct a two-distance set in a dimension. We survey the relationships between two-weight linear (n, k) codes over GF(q), projective (n, k, /»" /»,) sets in PG(£— \,q), and certain strongly regular graphs.

We also describe and. Strongly regular graphs. A graph X is strongly regular with parameters (n, k, λ, μ) if X has n vertices, every vertex has degree k, each pair of adjacent vertices has λ common neighbors, and each pair of non-adjacent vertices has μ common neighbors.

The complement of a SR graph is SR, so we may always assume k ≤ (n − 1) / 2. A Strongly Regular Graph Derived from the Perfect Ternary Golay Code 4.

Characterization Problems of Combinatorial Graph Theory 5. Line-Minimal Graphs with Cyclic Group 6. Circle Geometry in Higher Dimensions 7. Bose as Teacher—The Early Years 8. Construction of Symmetric Hadamard Matrices 9.

Cayley Diagrams and Regular Complex Polygons The question of whether a strongly regular graph with the above parameters is the graph of some partial geometry is of interest.

It was shown by Bose in that the answer is in the affirmative if a certain condition holds. Not much is known about the case if this condition is not satisfied, except for certain values of r and t.

of a partial geometry we call the strongly regular graph pseudo-geometric and if it. The non-existence of certain PBIB designs, Ann. Math. Statist., 30, –, Book. Jan   The chapter presents a more general theorem on complete bipartite induced sub graphs; most of the strongly regular graphs with ρ1= 3 contain no 3-claw.

With this, the proof of the standard form for the adjacency matrix of graphs and of the final theorems mainly is a.

The Petersen graph is strongly regular (with signature srg(10,3,0,1)). It is also symmetric, meaning that it is edge transitive and vertex transitive. More strongly, it is 3-arc-transitive: every directed three-edge path in the Petersen graph can be transformed into every other such path by a symmetry of the graph.

A strongly regular graph is defined as follows. Let G = (V,E) be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers λ and μ such that. Every two adjacent vertices have λ common neighbours.; Every two non-adjacent vertices have μ common neighbours.

A graph of this kind is sometimes said to be an srg(v, k, λ, μ). Strongly regular graphs lie on the cusp between highly structured and unstructured. For example, there is a unique strongly regular graph with parameters (36, 10, 4, 2), but there are non-isomorphic graphs with parameters (36, 15, 6, 6).

The goal of the present paper is to provide a gallery of small directed strongly regular graphs. For each graph of order n ≤ 12 and valency k strongly regular graphs is revealed, the full group of automorphisms is described, and some other nice properties are each graph a list of interesting subgraphs is provided as.InPenttila and Williford constructed an infinite new family of primitive Q-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space H (3, q 2), q even, with respect to a symplectic polar space W (3, q) embedded in it.

In a private communication to Penttila and Williford, H. Tanaka pointed out.Strongly regular graphs which have such a natural family of tight sets (of a certain type) include the polar spaces and the half dual polar graphs of diameter 2, see [10] and [11]. In fact, these.