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Integral Trees and Integral Graphs Ligong Wang Integral Trees and Integral Graphs Ligong Wang INTEGRAL TREES AND INTEGRAL GRAPHS c L. Wang, Enschede 2005. No part of this work may be reproduced by print, photography or any other means without the permission in writing from the author. Printed by W¨ohrmann Print Service, The Netherlands. ISBN: 90-365-2177-7 INTEGRAL TREES AND INTEGRAL GRAPHS PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus, prof. dr. W.H.M. Zijm, volgens besluit van het College voor Promoties in het openbaar te verdedigen op donderdag 16 juni 2005 om 15.00 uur door Ligong Wang geboren op 14 september 1968 te Qinghai, China Dit proefschrift is goedgekeurd door de promotoren prof. dr. Cornelis Hoede en prof. dr. Xueliang Li en assistent-promotor dr. Georg Still Contents Acknowledgement iii Preface v 1 Introduction 1 1.1 History of integral graphs and basic definitions . 2 1.1.1 Basicdefinitions ...................... 3 1.1.2 History of integral graphs . 6 1.2 Some formulae for the characteristic polynomials of graphs . 12 1.3 Surveyofresults .......................... 15 1.3.1 Resultsonintegraltrees . 15 1.3.2 Results on integral graphs . 28 1.3.3 Further results on integral graphs . 30 2 Some facts in number theory and matrix theory 31 2.1 Somefactsinnumbertheory . 31 2.1.1 Some specific useful results . 31 2.1.2 Some results on Diophantine equations . 33 2.2 Some notations from matrix theory . 38 3 Families of integral trees with diameters 4, 6 and 8 39 3.1 Integraltreeswithdiameter4 . 39 3.2 Integraltreeswithdiameters6and8 . 46 3.3 Furtherdiscussion ......................... 52 4 Integral trees with diameters 5, 6 and 8 55 4.1 Integraltreesofdiameter5 . 55 4.2 Two classes of integral trees of diameter 6 . 63 4.2.1 The characteristic polynomials of two classes of trees . 64 4.2.2 Integraltreesofdiameter6 . 65 i 4.3 Integraltreesofdiameter8 . 80 5 Integral complete r-partite graphs 85 5.1 A sufficient and necessary condition for complete r-partite graphs tobeintegral ............................ 85 5.2 Integral complete r-partitegraphs . 90 5.3 Furtherdiscussion ......................... 98 6 Integral nonregular bipartite graphs 99 6.1 The characteristic polynomials of some classes of graphs.... 99 6.2 Integral nonregular bipartite graphs . 108 6.3 Furtherdiscussion . 128 7 Families of integral graphs 131 7.1 Integral graphs K K and r K ............... 131 1,r • n ∗ n 7.2 Integral graphs K K and r K ............ 135 1,r • m,n ∗ m,n 8 Two classes of Laplacian integral and integral regular graphs141 8.1 The characteristic polynomials of two classes of regular graphs 141 8.2 Otherresults ............................ 144 Bibliography 153 Index 160 Summary 163 Curriculum vitae 165 Acknowledgement I would like to express my sincere gratefulness to many persons. Without their stimulation, cooperation and support, this thesis could not have been finished. Here I would like to mention some of them in particular. First of all I wish to express my deepest gratitude to my supervisors Prof. Dr. Cornelis Hoede and Prof. Dr. Xueliang Li. It was Prof. Dr. Xueliang Li who gave me the opportunity to be a Ph.D. student at the University of Twente. They have provided many useful suggestions and some new ideas when I discussed with them. Their ideas always inspired me to find new solutions to problems. Their stimulating enthusiasm and optimism created an excellent working atmosphere. It is a pleasure to work under their supervision. With Prof. Hoede’s warmhearted support, I spent a good time when I stayed in Twente. I would also like to thank my assistant-supervisor Dr. Georg Still. In the past four years, I stayed at the University of Twente for three times. During these stays I had many discussions with him. He also gave many useful suggestions and comments on an early version of this thesis. These suggestions helped me to considerably improve the presentation. At the same time, my life in Twente has been delightful due to his warm help and support. I am also grateful to Prof. Dr. Ir. H.J. Broersma, Prof. Dr. A.E. Brouwer, Prof. Dr. R. Martini and Prof. Dr. G.J. Woeginger for their willingness to participate in my graduation committee. I would also like to thank many colleagues, Prof. Dr. Ir. Hajo Broersma, Dr. Theo Driessen, Dr. Johann Hurink, Dr. Ir. Gerhard Post, Dr. Jan-Kees C.W. van Ommeren, Diny Heres-Ticheler, etc., in Twente. They also gave me much support in my research and my life. Thanks also go to the other Ph.D. students in the group, namely Xiaodong Liu, Shenggui Zhang, Lei Zhang, Hao Sun, Zhihui Li, Jichang Wu, Haixing Zhao and Xinhui Wang, for making my time of Ph.D. studies so en- joyable. During my study in Twente, I have shared an office with the Ph.D. students T. Brueggemann, P.S. Bonsma, A.N.M. Salman, G. Bouza and A.F. iii iv Acknowledgement Bumb. It was a memorable time to stay with them. Finally, I would also like to thank my parents, brothers and sisters. They provided me generous support and encouragement in these years, even though they did not always understand what I am doing. Last, but not least, I gratefully acknowledge my wife Xiaoyan Sun and my daughter Xian Wang for their support and love. Ligong Wang May 2005, Enschede Preface This thesis is the result of almost four years of research in the field of algebraic graph theory between September 2001 and March 2005. After an introductory chapter the readers will find seven chapters that contain four topics within this research field. These topics have, to varying extent, strong connections with each other. The first topic is on some facts in number theory and matrix theory. It is closely related to integral graphs or integral trees. The second topic deals with integral trees. The third topic is on integral graphs, cospectral graphs and cospectral integral graphs. The fourth topic deals with Laplacian integral and integral regular graphs. Some results of this thesis have been published in journals. See the following list: [1] L.G. Wang, X.L. Li and S.G. Zhang, Families of integral trees with diam- eters 4, 6 and 8, Discrete Appl. Math. 136 (2004), no.2-3, 349-362. [2] L.G. Wang, X.L. Li and C. Hoede, Integral complete r-partite Graphs, Discrete Math. 283 (2004), no.1-3, 231-241. [3] L.G. Wang, X.L. Li and C. Hoede, Two classes of integral regular graphs, Accepted for publication in Ars Combinatoria. v vi Preface Chapter 1 Introduction This thesis has four parts. The first part treats some facts in number theory and matrix theory. The second part is on integral trees. The third part deals with integral graphs, cospectral graphs and cospectral integral graphs. The fourth part is on Laplacian integral and integral regular graphs. The first part of the thesis consists of Chapter 2. In this part, we present several facts in number theory and matrix theory. The second part of the thesis consists of Chapters 3 to 4. In this part, some new families of integral trees with diameters 4, 5, 6 and 8 are characterized by making use of number theory and computer search. All these classes are infinite. They are different from those in the literature. We also prove that the problem of finding integral trees of diameters 4, 5, 6 and 8 is equivalent to the problem of solving Diophantine equations. This is a new contribution to the research of integral trees. We believe that it is useful for constructing other integral trees. In particular some special structures of integral trees of diameters 5, 6 and 8 are obtained for the first time. At the same time, some new results which treat interrelations between integral trees of various diameters are also found. These results generalize some well-known results or theorems on integral trees. The third part of the thesis consists of Chapters 5 to 7. In this part, firstly, we give a useful sufficient and necessary condition for complete r-partite graphs to be integral, from which we can construct infinitely many new classes of such integral graphs. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving some Diophantine equations. These results generalize Roitman’s result on the integral complete tripartite graphs. Secondly, fifteen classes of larger integral graphs are constructed from the known 21 smaller integral graphs. These classes consist of nonregular and 1 2 Chapter 1 bipartite graphs. Their spectra and characteristic polynomials are obtained from matrix theory. Their integral property is derived by using number theory and computer search. All these classes are infinite. These results generalize some results of Bali´nska and Simi´c. Thirdly, we determine the characteristic polynomials of four classes of graphs. We also obtain sufficient and necessary conditions for these graphs to be integral by using number theory and com- puter search. All these classes are infinite. We also give some new cospectral graphs and cospectral integral graphs. The fourth part of the thesis consists of Chapter 8. In this part, the spectra and characteristic polynomials of two classes of regular graphs are given. We also obtain the characteristic polynomials for their complement graphs, their line graphs, the complement graphs of their line graphs and the line graphs of their complement graphs. These graphs are not only integral but also Laplacian integral. These results generalize some results of Harary and Schwenk. We assume that the reader is familiar with the essentials of graph theory. Most of the terminology and notations can be found in Bondy & Murty [10], Cvetkovi´c, Doob & Sachs [22] or Harary [35].
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