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Graphs with Few Eigenvalues Graphs with few eigenvalues An interplay between combinatorics and algebra Graphs with few eigenvalues An interplay between combinatorics and algebra Proefschrift ter verkrijging van de graad van doctor aan de Katholieke Universiteit Brabant, op gezag van de rector magnificus, prof. dr. L.F.W. de Klerk, in het openbaar te verdedigen ten overstaan van een door het college van dekanen aangewezen commissie in de aula van de Universiteit op vrijdag 4 oktober 1996 om 14.15 uur door Edwin Robert van Dam geboren op 27 augustus 1968 te Valkenswaard Promotor: Prof. dr. S.H. Tijs Copromotor: Dr. ir. W.H. Haemers Ter nagedachtenis van Opa Vogel Acknowledgements During the past four years I have had the pleasure to work on this thesis under the supervision of Willem Haemers. His support and guidance have always been stimulating, and I have found Willem always available. So thanks Willem, I had a great time. I also wish to thank Stef Tijs, for being my promoter, and the other members of the thesis committee: Andries Brouwer, Giel Paardekooper, Lex Schrijver and Ted Spence. Special thanks go to Ted Spence for the enjoyable cooperation in the past years, and for always providing me with interesting combinatorial data. Furthermore I would like to thank Jaap Seidel for inspiring me, D.G. Higman for his inspiring manuscript on rank four association schemes, and all other people who showed interest in my work. During the past years I attended several graduate courses with great pleasure, and I thank the organizers and lecturers of LNMB and EIDMA for those. I also gratefully acknowledge CentER for publishing this thesis. This summer I visited the University of Winnipeg and the University of Waterloo, Canada. These visits helped me to finish the last details of this thesis, and have given me lots of inspiration. Accordingly I wish to thank Bill Martin and Chris Godsil for the kind invitations. Many thanks go to my office-mate Paul Smit for too many things to write down. Also thank you to my nearest colleagues, and to the secretaries for making Tilburg University a nice place to stay. Finally I would like to thank my parents for always helping out, and my dearest friend of all, Inge van Breemen, for her beautiful sunshine, which is reflected in this thesis. Tilburg, August 1996 Edwin van Dam Contents Let me explain to you, Socrates, the order in which we have arranged our entertainment. (Plato, Timaeus) Acknowledgements ............................................... vii Contents ...................................................... ix 1. Introduction .................................................. 1 1.1. Graphs, eigenvalues and applications .............................. 1 1.2. Graphs with few eigenvalues - A summary ......................... 3 1.3. Graphs, combinatorics and algebra - Preliminaries ..................... 4 1.3.1. Graphs .............................................. 4 1.3.2. Spectra of graphs ....................................... 5 1.3.3. Interlacing ............................................ 9 1.3.4. Graphs with least eigenvalue −2 ............................. 10 1.3.5. Strongly regular graphs ................................... 11 1.3.6. Association schemes ..................................... 12 1.3.7. Designs and codes ...................................... 16 1.3.8. Switching and the Seidel spectrum ........................... 17 1.3.9. A touch of flavour - Graphs with few Seidel eigenvalues ........... 17 2. Graphs with three eigenvalues .................................... 19 2.1. The adjacency spectrum ....................................... 19 2.1.1. Nonintegral eigenvalues ................................... 21 2.1.2. The Perron-Frobenius eigenvector ............................ 23 2.1.3. Graphs with least eigenvalue −2 ............................. 24 2.1.4. The small cases ........................................ 29 2.2. The Laplace spectrum - Graphs with constant µ and µ .................. 31 2.2.1. The number of common neighbours and vertex degrees ............. 32 2.2.2. Cocliques ............................................. 36 2.2.3. Geodetic graphs of diameter two ............................ 38 2.2.4. Symmetric designs with a polarity ........................... 39 2.2.5. Other graphs from symmetric designs. ......................... 41 2.2.6. Switching in strongly regular graphs .......................... 42 x Contents 2.3. Yet another spectrum ......................................... 43 3. Regular graphs with four eigenvalues ............................... 45 3.1. Properties of the eigenvalues ................................... 46 3.2. Walk-regular graphs and feasibility conditions ....................... 47 3.2.1. Feasibility conditions and feasible spectra ...................... 47 3.2.2. Simple eigenvalues ...................................... 49 3.2.3. A useful idea .......................................... 51 3.3. Examples, constructions and characterizations ........................ 51 3.3.1. Distance-regular graphs and association schemes ................. 51 3.3.2. Bipartite graphs ........................................ 52 3.3.3. The complement of the union of strongly regular graphs ............ 52 3.3.4. Product constructions .................................... 52 3.3.5. Line graphs and other graphs with least eigenvalue −2 ............. 57 3.3.6. Other graphs from strongly regular graphs ...................... 59 3.3.7. Covers ............................................... 64 3.4. Nonexistence results ......................................... 67 4. Three-class association schemes .................................... 73 4.1. Examples ................................................. 73 4.1.1. The disjoint union of strongly regular graphs .................... 74 4.1.2. A product construction from strongly regular graphs ............... 75 4.1.3. Pseudocyclic schemes .................................... 75 4.1.4. The block scheme of designs ............................... 76 4.1.5. Distance schemes and coset schemes of codes ................... 76 4.1.6. Quadrics in projective geometries ............................ 77 4.1.7. Merging classes ........................................ 77 4.2. Amorphic three-class association schemes .......................... 78 4.3. Regular graphs with four eigenvalues ............................. 80 4.3.1. The second subconstituent of a strongly regular graph .............. 81 4.3.2. Hoffman cocliques in strongly regular graphs .................... 83 4.3.3. A characterization in terms of the spectrum ..................... 83 4.3.4. Hoffman colorings and systems of linked symmetric designs ......... 90 4.4. Number theoretic conditions .................................... 93 4.4.1 The Hilbert norm residue symbol and the Hasse-Minkowski invariant . 93 4.5. Lists of small feasible parameter sets .............................. 95 5. Bounds on the diameter and special subsets ........................... 97 5.1. The tool .................................................. 98 5.2. The diameter .............................................. 99 5.2.1. Bipartite biregular graphs ................................. 102 5.2.2. The covering radius of error-correcting codes ................... 104 Contents xi 5.3. Bounds on special subsets .................................... 106 5.3.1. The number of vertices at maximal distance and distance two ....... 108 5.3.2. Special subsets in graphs with four eigenvalues ................. 110 5.3.3. Equally large sets at maximal distance ....................... 114 Appendices ................................................... 117 A.2. Graphs with three Laplace eigenvalues ........................... 117 A.3. Regular graphs with four eigenvalues ............................ 119 A.4. Three-class association schemes ................................ 123 Bibliography .................................................. 135 Index ........................................................ 141 Samenvatting .................................................. 143 Hooggeleerde heer, ik heb deze proef zelf genomen en ook zelf de metingen verricht. Ik ben op een nacht naar een graanmolen geslopen met een graankorrel in de hand. Die graankorrel heb ik op de onderste molensteen gelegd. Toen heb ik met een zware takel de tweede molensteen op de graankorrel laten zakken, zodat ik een stenen sandwich had verkregen: twee molenstenen met een graankorrel ertussen. Nu heeft een graankorrel de merkwaardige eigenschap dat zodra hij tussen twee stenen wordt geklemd hij de neiging krijgt om de losliggende steen, en dat is altijd de bovenste, in beweging te willen brengen. De graankorrel wil zich wentelen, hij wil als het ware gemalen worden. Meel! wil hij worden. Maar dat gaat zomaar niet. Daartoe moet hij zich verschrikkelijk inspannen om één van die stenen, de bovenste, te laten draaien. Maar op de lange duur heeft de korrel zich al een paar centimeter voortgewenteld. En dan gaat het proces steeds sneller en alléén maar door de wil en de macht van de graankorrel moet de bovenliggende molensteen de zware molenas in beweging brengen en door diezelfde wil worden de raderen in de kop van de molen in beweging gebracht, de as die naar buiten steekt en de wieken die eraan bevestigd zijn. Zo maken molens evenals ventilators wind. Met dit verschil dat de laatste op elektriciteit lopen en de eerste gewoon op graankorrels. U kunt zeker wel begrijpen hoe hard het gaat waaien als je een paar duizend korrels tegelijk gebruikt ? ... Mijne heren ... de enige die
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