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University Microfilms International 300 North Zeeb Road Ann Arbor, Michigan 48106 USA St. John's Road, Tyler's Green High Wycombe, Bucks, England HP10 8HR 77-24,677 NEMZER, Daniel Edward, 1947- CHARACTERIZATION OF LINE GRAPHS. The Ohio State University, Ph.D., 1977 Mathematics Xerox University Microfilms, Ann Arbor, Michigan 4Bioe CHARACTERIZATION OF LINE GRAPHS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Daniel Edward Nemzer, B.S., M.S. ***** The Ohio State University 19 77 Reading Committee* Approved By Professor Dijen K. Ray-Chaudhuri t to s'* nf Professor Thomas A. Dowling • 7 — ( k.. Professor G. Neil Robertson Adviser Professor Richard M. Wilson Department of Mathematics ACKNOWLEDGMENTS I am deeply grateful to my adviser, Prof. Dijen K. Ray-Chaudhuri. He guided me in the research reported in this dissertation. The members of the reading committee made many helpful suggestions. Prof. Arnold E. Ross (at The Ohio State University), and Prof. Frederick B. Thompson (at the California Institute of Technology), are among the teachers who have inspired my studies. Mrs. E. L. Karlquist, of the OSU Mathematics Library, helped me to locate many useful journal articles. This dissertation is dedicated to my mother, Mrs. Louis Nemzer, and to the memory of my father, the late Prof. Louis. Nemzer. The loving guidance of my parents has been a blessing. Finally, I thank my wife, Elaine. She has been a constant source of encouragement and understanding. VITA July 8, 19^7 • • Born— Washington, D.C. 1969 ........... B.S. with Honor, California Institute of Technology, Pasadena, California 1969-1971 • • • University Fellow, The Ohio State University, Columbus, Ohio 1970 • . M.Si,' The Ohio State University, Columbus, Ohio 1970-1971 • • • Graduate Teaching Associate, Department of Mathematics, The Ohio State University 1971-1973 • • • Mathematician and Computer Analyst, U.S. Public Health Service, Rockville, Maryland 1973-197^ • • • Graduate Teaching Associate, Department of Mathematics, The Ohio State University 197^f ...... Graduate Research Associate, Department of Mathematics, The Ohio State University 1973-1975 • • • University Fellow, The Ohio State University, Columbus, Ohio FIELDS OF STUDY Major Field* Mathematics Studies in Combinatorics and Graph Theory* Professors D. K. Ray-Chaudhuri, Thomas A* Dowling, G. Neil Robertson, and Richard M. Wilson Studies in Algebra* Professor Harold D. Brown Studies in Number Theory* Professor Arnold E. Ross iii TABLE OP CONTENTS Page ACKNOWLEDGMENTS ..................................... ii VITA ..................................................... iii LIST OP T A B L E S .......... vi LIST OF FIGURES ........................................ vii INTRODUCTION .......................................... 1 Chapter I. DEFINITIONS ................................. 11 1• Graphs 2. The line graph of a graph 3* The line graph of a tree 4. The eigenvalues of a graph 5. Literature review II. THE LINE GRAPH OF A BALANCED INCOMPLETE BLOCK DESIGN ................................... 30 1. Introduction 2. The line graph of a semiregular bigraph 3. Nonedge degree at most 1 4. The parameters of a BIBD 5* No subgraph K^-Kg 6 . The line graph of a BIBD III. GENERALIZED LINE G R A P H S ...................... 49 1. Introduction 2. Cocktail-party graphs 3* Generalized line graphs 4. Permitted graphs 5• Forbidden graphs IV. GRAPHS WITH LEAST EIGENVALUE £ - 2 ........... 67 1. Introduction 2. The edge-signed graph of inner products 3. The base graph of a set of vectors 4. Root systems 5* The base graph of a graph V. CONSTRUCTION OF EXCEPTIONAL GRAPHS ........... 100 1. Introduction 2. Construction from a base graph 3. Exceptional trees 4. Graphs represented in E^ 5• Exceptional graphs VI. THE LINE MULTIGRAPH OF A M U L T I G R A P H .......125 1. Introduction 2. Multigraphs 3. The line raultigraph 4. Geometric characterizations 5. Multigraphs with least eigenvalue ^ -2 APPENDICES A. THE 143 CONNECTED GRAPHS ON 1 TO 6 VERTICES . 144 B. DIAGRAMS OF REGULAR EXCEPTIONAL GRAPHS .... 149 C. DIAGRAMS OF NON-REGULAR EXCEPTIONAL GRAPHS . 156 BIBLIOGRAPHY .......................................... 161 v LIST OF TABLES Chapter I Page Table 5 • 1 ■ Characterizations of graphs 26 Chapter II Table 1.1. Geometric characterizations of the line graph of a BIBD 31 Table 4.1. The line graph of a nontrivial, nonsymmetric BIBD 42 Chapter IV Table 3.1. The base graph of a set S of vectors 78 Table 5-1. The base graph of a graph H 93 Chapter V Table 4.1. The 42 patterns of X, Y, W in 113 Table 4.2. An upper bound on the number of exceptional graphs in E^ 119 Appendix A Table A.I. The number of graphs on 1 to 6 vertices 144 Table A.2. Eigenvalues of the 143 connected graphs on 1 to 6 vertices 146 Appendix B Table B .1. The 187 regular connected exceptional graphs 150 LIST OF FIGURES Chapter I Page Fig. 1.1. The graphs Kj, K^, K^-Kg, and ^ 12 Fig. 2.1. Ten connected graphs and their line graphs 15 Fig. 2.2. Odd and even triangles 16 Fig. 2.3* The nine graphs forbidden in a line graph 18 Fig. 3.1. An even triangle in H = L(G) 22 Fig. if-.l. Two isospectral graphs 2k Fig. 5•1• The two Kuratowski graphs, and ^ 29 Chapter II Fig. 3.1. Two graphs which are not line graphs 39 Fig. 5-1. The graph K k-K ^3 Fig. 5.2. A four-cycle in the line graph of a BIBD k5 Chapter III Fig. 3*1* A graph and a generalized line graph 52 Fig. k.l. j in two generalized line graphs 55 Fig• k .2. Ktj-Kg in four generalized line graphs 56 Fig. ^.3. The graph 57 Fig. k,k* The graph 59 Fig. 5•1• The nine graphs forbidden in a line graph 60 Fig. 5.2. K^-Kg in a generalized line graph 61 Fig. 5*3* K^-Kg not in a generalized line graph 61 Fig. 5-k. Gg not in a generalized line graph 62 Fig. 5.5. G^ not in a generalized line graph 63 Fig. 5.6. G- not in a generalized line graph 64 Fig. 5.7. Gg not in a generalized line graph 65 Fig. 5*8. Gg not in a generalized line graph 68 Chapter IV Fig. 2 .1. Three adjacent vectors in a star or cone 71 Fig. 2 .2 . The edge-signed graph of a,b,c,x^ 72 Fig. 2.3. The edge-signed graph of a,b,c,y2 72 Fig. 2.4. The edge-signed graph of a fbtc(z^ 73 Fig. 2.5. The edge-signed graph of a,b,c,x^,3/ri,zi 74 Fig. 2 .6 . Type 1 connection, with (x^»x^) - 1 75 Fig. 2.7. Type 0 connection, with (x^,x2) = 0 76 Fig. 3.1. The line graph L(K^ 80 Fig. 3.2. The line graph Xj(K^5 81 Fig. 3 0 . Segre coordinates for the Schldfli graph 82 Fig. 3.4. Inner products of Cameron et al. (1978) 83 Fig. 4.1. The graphs Q, y and K^-LI 90 Fig. 5.1. H does not contain K2 ^ or K^-LI 91 Fig. 5.2. Type 1 connection, with (xi»xj) = 1 94 Fig. 5.3. Type 0 connection, with (x^.x.) - 0 94 Fig. 5.4. H does not contain Q or K^-1I 97 Chapter V Fig. 3.1. Cycles in H are not permitted 105 Fig. 3.2. The six base graphs of exceptional trees 106 Fig. 3*3* The six T vertices in Eg 107 Fig. 3.4. The two exceptional graphs in Eg 107 Fig. 3-5- Signs for T in , and the graph H 109 • • a V l l l Pig. 3.6. Signs for T in B^, and the graph H 110 Fig. 4.1. Substitutions for Pattern 1 118 Pig.