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Graphs and Semibiplanes by Jennifer A. Muskovin On (0,2)-graphs and Semibiplanes by Jennifer A. Muskovin (Under the direction of Lenny Chastkofsky) Abstract A semibiplane is a connected point-block incidence structure such that any two points are in either 0 or 2 common blocks and any two blocks have either 0 or 2 common neigh- bors. A (0,2)-graph is a connected graph such that any pair of vertices has either 0 or 2 common neighbors. The incidence graph of a semibiplane is a bipartite (0,2)-graph. In this paper we construct (0,2)-graphs from known graphs by taking Cartesian products, quotients and adding matchings and construct semibiplanes by taking quotients of finite projective planes. Index words: (0,2)-graph, Biplane, Hypercube, Projective Plane, Semibiplane On (0,2)-graphs and Semibiplanes by Jennifer A. Muskovin B.S., Benedictine University, 2007 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Master of Arts Athens, Georgia 2009 c 2009 Jennifer A. Muskovin All Rights Reserved On (0,2)-graphs and Semibiplanes by Jennifer A. Muskovin Approved: Major Professor: Lenny Chastkofsky Committee: Dino Lorenzini Robert Varley Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia May 2009 Dedication To Mom and Papa iv Acknowledgments I would like to thank the faculty, staff and students within the Mathematics Department at UGA for encouraging me to finish writing. Specifically, thank you Lenny for giving me so much of your time this past year and for staying patient with me during my many struggles. I would also like to thank Joe for putting so much work into writing his paper. I admit that at times my motivation was knowing that you were consistently working until 2 a.m. Finally, thank you to Matt and Whitney for making me smile so many times this year. v Table of Contents Page Acknowledgments . v List of Figures . viii List of Tables . ix Chapter 1 Introduction and Background . 1 1.1 Graph Definitions and Examples . 1 1.2 Design Definitions and Examples . 4 2 Graph Constructions . 7 2.1 General Results of (0,2)-graphs . 7 2.2 Semibiplanes and (0,2)-graphs . 8 2.3 Cartesian Product of Graphs . 11 2.4 Quotient Graphs . 13 2.5 Bipartite Double . 16 2.6 Matchings . 18 3 Biplanes . 20 3.1 Existence of Symmetric Block Designs . 20 3.2 Existence of Biplanes . 22 4 Constructing Semibiplanes from Finite Projective Planes . 26 4.1 Collineations of a Finite Projective Plane . 28 4.2 Semibiplane Construction . 29 vi vii 4.3 Using GAP to Investigate PG(n,q) . 31 5 Hypercubes . 37 5.1 The Automorphism Group of Qk ................ 37 5.2 Quotients of Hypercubes Resulting in Nonbipartite Graphs 38 5.3 Quotients of Folded Qk ..................... 40 5.4 Quotients of coHeawood ×Qk .................. 43 Appendix A Code Used to Find “Good Conjugacy” Class Relations . 47 B Code to Construct Graphs of CoHeawood × Qk .......... 50 C Code to Construct SemiBiplane and Incidence Graph from Finite Projective Plane . 53 D Sample List of Bipartite Graphs from [2] ............... 55 Bibliography . 56 List of Figures 1.1 The octahedral graph with colored antipodal vertices . 2 1.2 Petersen Graph . 3 1.3 The Heawood Graph and coHeawood Graph . 4 1.4 The coHeawood Graph . 5 2.1 Given u, v, w in a (0,2)-graph we can find a unique y completing the quadrangle. 8 ∼ 2.2 K2 × Q2 = Q3 ................................... 12 2.3 Q3 and K4 ..................................... 14 2.4 Bipartite Double of Q2 .............................. 17 2.5 Q3 and K4 × K2 ................................. 19 4.1 Projective Plane of order n = 2: Fano Plane . 27 4.2 An Elation of the Fano Plane . 28 4.3 A Homology of a Projective Plane of Order 3 . 29 4.4 Constructing the incidence graph of the square from the Fano Plane . 31 5.1 Quotients of Q6 .................................. 41 5.2 Shrikhande graph with code labels . 42 viii List of Tables 3.1 Using Theorem 3.1.1 to find v given k, supposing such a design existed . 22 3.2 Known biplanes from [6]; * indicates the dual of a biplane . 24 4.1 Points and Lines of the Fano Plane . 27 4.2 Points and Lines of PG(2,4) . 32 4.3 Points and Blocks of the semibiplane . 33 4.4 Semibiplanes Constructed from Finite Projective Planes . 36 5.1 An Automorphism of Q3 ............................. 37 5.2 Graphs Isomorphic to Quotients of Λ × Qk, k = 2, 3, 4............. 46 D.1 ∆0.1 − ∆6.4 from [2] . 55 ix Chapter 1 Introduction and Background 1.1 Graph Definitions and Examples In this paper, we will assume that all graphs are finite, connected, simple undirected graphs. Unless otherwise specified we will use ∼ to denote adjacency. We will use Qk to denote the graph of the k-hypercube with valency k, V (Γ) and E(Γ) to denote the vertex set and edge set of a graph Γ and ∂(u, v) to denote the distance between vertices u and v. The remainder of this section contains the definitions we will need in the following chapters. A (0,2)-graph Γ is a connected graph with the property that any pair of vertices has 0 or 2 common neighbors. A (0,2)-graph without any triangles is called a rectagraph. In other words a rectagraph is a connected triangle-free graph with the property that every path of length two is in a unique quadrangle. Notice that a bipartite (0,2)-graph is a rectagraph. If Γ is a bipartite (0,2)-graph then it cannot contain any triangles. If it did, then it would force an adjacency between 2 vertices of the same partition of the vertex set. In particular, Qk is a rectagraph. An isomorphism between the graphs Γ and Γ0 is a bijection f : V (Γ) → V (Γ0) such that any two vertices u, v ∈ Γ are adjacent in Γ if and only if f(u) and f(v) are adjacent in Γ0. An isomorphism from Γ to itself is an automorphism of Γ. In other words, an automorphism of a simple graph Γ is just a permutation of its vertices with the same adjacency property. The set of all automorphisms of Γ along with the operation of composition forms a group called the automorphism group of Γ, denoted Aut(Γ). For some graphs, especially those of the hypercubes, it will be useful to label the vertices as binary code words or vectors. Let S=F2 and k ∈ N.A binary code of length k over S is a 1 2 Figure 1.1: The octahedral graph with colored antipodal vertices subset of Sk. The elements of the code are called code words and the support of a code word u is {i : ui 6= 0}. The distance between two code words is given by the number of positions in which the two differ. For instance, the distance between (0,1,0) and (1,1,1) is 2 while the distance between (0,1,0) and (1,0,1) is 3. We can alternatively define Qk as the graph with vertices all binary codes words of length 2k with an edge connecting two vertices when the distance between their respective code words is exactly 1. When the distance between two length k code words is k we say the code words are antipodal. The antipodal graph of a graph Γ has the same vertex set as Γ with an edge connecting u, v ∈ V (Γ) if the distance between u and v in Γ is equal to the diameter of Γ. We say a graph Γ with diameter d is antipodal if for any vertices u, v, w such that ∂(u, v) = ∂(u, w) = d either ∂(v, w) = d or v = w. The graphs of the hypercubes are antipodal graphs because every vertex has a unique vertex at a maximum distance from it. Example: The octahedral graph as pictured in Figure 1.1 is an example of an antipodal graph. The red vertices are antipodal as are the blue and green vertices. Let Γ be a graph with u ∈ V (Γ) we will use Γi(u) to denote the set of vertices v such that ∂(v, u) = i. A connected graph Γ is called distance regular if there are integers bi, ci 3 Figure 1.2: Petersen Graph (i ≥ 0) such that for any two vertices u, v of Γ at distance i from each other, there are precisely ci neighbors of v in Γi−1(u) and bi neighbors of v in Γi+1(u). The intersection array of a distance regular graph Γ is the sequence ι(Γ) := {b0, b1, ..., bd−1; c1, c2, ..., cd}. Note that the numbers in the intersection array are independent of u, v. Further, Γ is k − regular with k = b0 as b0 is by definition the number of vertices adjacent to a given vertex. It will also always be the case that bd = 0 because Γd+1(u) will be empty. Similarly, c0 is undefined as Γ−1(u) is empty. Clearly, Γ0(u) = 1 as u can only be of distance 0 away from itself so if ∂(u, v) = 1 then the number of neighbors of v in Γ0(u) is also 1 so c1 = 1. Example : Let Γ be the Petersen graph as labeled in Figure 1.2. The diameter of Γ is 3. Let u = 1. Then Γ1(u) = {2, 6, 5} and Γ2(u) = {3, 4, 7, 8, 9, 10}. Let v = 2 then ∂(v, u) = 1 0 0 so b1 = 2 as 2 ∼ 3, 7.
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