INVESTIGATIONS INTO THE RANKS OF REGULAR GRAPHS by CHARLES R GARNER, JR DISSERTATION submitted in the fulfilment of the requirements for the degree PHILOSOPHIAE DOCTOR in MATHEMATICS in the FACULTY OF SCIENCE at the RAND AFRIKAANS UNIVERSITY PROMOTER: DR E JONCK MARCH 2004 INVESTIGATIONS INTO THE RANKS OF REGULAR GRAPHS CHARLES R GARNER, JR. I hearby declare that the dissertation submitted for the Philosophiae Doctor degree to the Rand Afrikaans University, apart from the help recognised, is my own work and has not been formerly submitted to another university for a degree. Charles R Garner Jr ACKNOWLEDGEMENTS I wish to express my heartfelt gratitude and appreciation to my supportive family: my wife and my two wonderful sons. I wish to thank my parents for their support throughout my life in any endeavour I chose to pursue. I wish to thank Dr. Gayla S. Domke for her amazing support and encouragement as I worked on my master's thesis and this doctoral dissertation. Her support has been instrumental in my pursuit of both my master's and doctoral degrees. I wish to thank Dr. George J. Davis for his incredible acceptance of me as a research colleague and his willingness to also support my pursuit of a doctoral degree. I wish to express thanks also to Dr. Elizabeth Jonck for her encouragement, support, and advice throughout this journey towards a doctoral degree. My goal could not have been achieved without her. Finally, I wish to thank the external examiners. They provided useful references and suggestions that helped to make this work better. Contents 0.1 Summary 1 0.2 Opsomming 3 1 Introduction 5 2 Preliminary Results 7 2.1 Results from Matrix Theory 7 2.2 Ranks of Regular Graphs 9 3 Strongly Regular Graphs 14 3.1 Rank of Strongly Regular Graphs 14 3.2 Examples of Strongly Regular Graphs 17 4 Ranks of Graph Products of Regular Graphs 23 4.1 Cartesian Products 24 4.2 Complete Products 35 5 Ranks of Line Graphs 39 5.1 Line Graphs of Regular Graphs 40 5.2 Line Graphs of Strongly Regular Graphs 50 5.3 Line Graphs of Regular Graph Products 53 6 Ranks of Complements of Regular Graphs 60 6.1 Complements of Regular Graphs 61 6.2 Complements of Strongly Regular Graphs 63 6.3 Complements of Regular Graph Products 67 7 Ranks of Regular Graphs Under Other Unary Operations 71 7.1 The Subdivision 75 7.2 The Connected Cycle 82 7.3 The Complete Subdivision 86 7.4 The Total Graph 90 8 Ranks of Graphs Involving Paths 98 8.1 Paths and Cartesian Products Involving Paths 98 8.2 The Line Graph, the Complement, and Other Unary Operations on Paths 99 9 Conclusion 106 References 107 1 0.1 Summary Subjects: Graph Theory, Spectral Theory, Eigenvalues In this thesis we investigate the ranks of many classes of regular graphs. We also discuss the structure of strongly regular graphs and graphs under certain unary and binary operations. In Chapter 1 we discuss the rationale behind investigating ranks of graphs. In Chapter 2 we give relevant definitions and results from matrix theory. We also summarise existing results concerning ranks of regular graphs. In Chapter 3 we present the structural properties of strongly regular graphs and the relationship these properties have to the spectrum of a strongly regular graph. We then determine the rank of a general strongly regular graph, followed by the ranks of specific types of strongly regular graphs. In Chapter 4 we investigate the structure of two types of binary graph products: the Cartesian product and the complete product. We discuss the transformation the spectra of two regular graphs undergo to form the spectra of the product. We then determine the ranks of these products on regular graphs by examining their spectra. In Chapter 5 we begin the investigation of the ranks of graphs under unary operations. This chapter examines the transformation of the spectrum of a regular graph under the 2 line graph operation. The ranks of the line graphs of many regular graphs are then determined. In Chapter 6 we continue with the unary operations by discussing the ranks of the complements of regular graphs. Here, we again examine the transformation of the spectrum to determine the rank. In Chapter 7 we conclude the exploration of unary operations by investigating the subdivision graph, the connected cycle graph, the connected subdivision graph, and the total graph. Here, the bipartite property plays a major role in the transformation of the spectrum of a regular graph under these operations; thus, we include relevant results concerning the rank of regular bipartite and regular semi-bipartite graphs. In Chapter 8 we investigate the rank of a special non-regular graph, the path. We determine ranks of graph products involving paths and unary operations on paths by examining both structural properties and spectrum transformations. Chapter 9 concludes the dissertation by summarising the previous work and pointing the way towards future directions for research. 3 0.2 Opsomming Onderwerpe: Grafiekteorie, Spektraalteorie, Eiewaardes In hierdie proefskrif ondersoek ons die rang van baie klasse van reguliere grafieke. Ons bespreek die struktuur van sterk reguliere grafieke en grafieke onder sekere unere en binere bewerkings. In Hoofstuk 1 bespreek ons die sin van die bestudering van die rang van grafieke. In Hoofstuk 2 word relevante definisies en resultate van matriksteorie bespreek. Ons som ook bestaande resultate in verband met die rang van reguliere grafieke op. In Hoofstuk 3 gee ons die strukturele eienskappe van sterk reguliere grafieke weer, asook die verwantskap wat hierdie eienskappe met die spektrum van 'n sterk reguliere grafiek het. Daarna bepaal ons die rang van 'n sterk reguliere grafiek in die algemeen, gevolg deur die rang van spesifieke tipes sterk reguliere grafieke. In Hoofstuk 4 ondersoek ons die stuktuur van twee tipes binere grafiekprodukte: die Cartesiese produk en die volledige produk. Ons bespreek die transformasie wat die spektrums van twee reguliere grafieke ondergaan om die spektrum van die produk te vorm. Daarna bepaal ons die rang van hierdie produkte van reguliere grafieke deur die spektrum daarvan te ondersoek. 4 In Hoofstuk 5 begin ons met die ondersoek van die rang van grafieke onder unere bewerkings. Hierdie hoofstuk ondersoek die transformasie van die spektrum van 'n reguliere grafiek onder die lyngrafiekoperasie. Die rang van die lyngrafiek van baie reguliere grafieke word dan bepaal. In Hoofstuk 6 word unere bewerkings weer gebruik. Die rang van die komplemente van grafieke geniet aandag. Ook hier word die transformasie van die spektrum ondersoek om die rang te bepaal. In Hoofstuk 7 voltooi ons die ondersoek van unere bewerkings deur die subverdelingsgrafiek, die samehangende subverdelingsgrafiek en die totale grafiek te beskou. Hier speel die tweeledigheidseienskap 'n groot rol in die transformasie van die spektrum van 'n reguliere grafiek onder genoemde bewerkings; daarom sluit ons relevante resultate omtrent die rang van reguliere tweeledige en reguliere semi-tweeledige grafieke in. In Hoofstuk 8 ondersoek ons die rang van 'n spesiale nie-reguliere grafiek, naamlik die pad. Ons bepaal die rang van grafiekprodukte wat paaie betrek en unere bewerkings op paaie deur beide strukturele eienskappe en spektrumtransformasies te bestudeer. Hoofstuk 9 sluit die proefskrif of deur 'n opsomming te maak van voorafgaande werk en die pad na toekomstige navorsing aan te dui. 5 1 Introduction The present work concerns properties of graphs. An undirected graph G = (V, E) with no loops or multiple edges is a finite set V = {1, 2, ..., n} of elements called vertices together with a set E consisting of two-element subsets of V, called edges. Much recent research has focused on relating properties of the adjacency matrix of a graph to the underlying graph. An adjacency matrix of a graph is a matrix describing which vertices are adjacent. Two vertices are adjacent if they are connected by an edge; i. e., if j} E E for vertices i and j. The adjacency matrix, A(G), of a graph G on n vertices is an n x n matrix consisting of entries ay = k, where k is the number of edges connecting vertices i and j. An example of a graph and its adjacency matrix is shown in Figure 1. Since we only consider graphs with no loops or multiple edges in this thesis, all entries in the adjacency matrix are either 0 or 1. One such property of interest is the rank of the adjacency matrix, also called the rank of the graph. The rank of a graph G, denoted rank(G), has been found to be the upper bound for numerous graph parameters. This importance of the rank, due to applications in physics, chemistry and combinatorics, has spurred work in the determination of the rank for many types of graphs. Previous work includes ranks of trees, grid graphs [BDM], and circulants [DD, DDG1]; ranks of graphs after vertex addition [BBD2]; and, ranks of graphs after edge insertion or deletion [DavG]. 6 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 0 2 0 0 0 2 0_ Figure 1. A graph and its adjacency matrix In this thesis, the ranks of many types of regular and strongly regular graphs are determined. Also determined are ranks of regular graphs under unary operations: the line graph, the complement, the subdivision graph, the connected cycle, the complete subdivision graph, and the total graph. The binary operations considered are the Cartesian product and the complete product. The ranks of the Cartesian product of regular graphs have been investigated previously in [BBD1]; here, we summarise and extend those results to include more regular graphs.