| Acknowledgements | 第7-8页 |
| Abstract | 第8-9页 |
| 1 Introduction | 第12-24页 |
| 1.1 Root latices and the corresponding graphs | 第13-17页 |
| 1.1.1 Hoffman problem | 第14页 |
| 1.1.2 Exceptional graphs with smallest eigenvalue more than-2 | 第14-16页 |
| 1.1.3 Generalized line graphs | 第16-17页 |
| 1.2 Graphs with smallest eigenvalue less than-2 | 第17-19页 |
| 1.2.1 Hoffman graphs with smallest eigenvalue at least-3 | 第18-19页 |
| 1.3 Signed graphs and root lattices | 第19-20页 |
| 1.4 Integrability of graphs | 第20-24页 |
| 1.4.1 A lattice related to a graph | 第21-24页 |
| 2 Graphs, Hoffman graphs and lattices | 第24-42页 |
| 2.1 Graphs | 第24-29页 |
| 2.1.1 Strongly regular graphs | 第27-28页 |
| 2.1.2 Eigenvalues of a graph | 第28-29页 |
| 2.2 Hoffman graphs | 第29-32页 |
| 2.3 Lattices | 第32-42页 |
| 2.3.1 Irreducible root lattices | 第33-35页 |
| 2.3.2 s-integrability of lattices | 第35-36页 |
| 2.3.3 Graphs and lattices | 第36-39页 |
| 2.3.4 Hoffman graphs and lattices | 第39-42页 |
| 3 The integrally representable trees of norm 3 | 第42-58页 |
| 3.1 Some results on an integrally representable Hoffman graph (?) of norm 3 | 第42-43页 |
| 3.2 Tree-like Hoffman graphs | 第43-49页 |
| 3.2.1 Stripped Hoffman graphs | 第43-44页 |
| 3.2.2 Some results on tree-like Hoffman graphs | 第44-47页 |
| 3.2.3 A family of(-3)-irreducible tree-like Hoffman graphs | 第47-49页 |
| 3.3 Integrally representable tree-like Hoffman graphs | 第49-55页 |
| 3.4 Seedlings | 第55-58页 |
| 4 On the integrability of strongly regular graphs | 第58-76页 |
| 4.1 Equitable partition and quotient matrix | 第59-60页 |
| 4.1.1 Designs | 第59-60页 |
| 4.2 Basic results on s-integrable graphs | 第60-62页 |
| 4.3 Integrable strongly regular graphs | 第62-65页 |
| 4.3.1 Two classes of integrable strongly regular graphs | 第62-63页 |
| 4.3.2 Integrable strongly regular graphs with smallest eigenvalue at least-2 | 第63-64页 |
| 4.3.3 Integrable strongly regular graphs with smallest eigenvalue at least-3 | 第64-65页 |
| 4.4 The complement of the Sims-Gewirtz graph | 第65-66页 |
| 4.5 The Hoffman-Singleton graph | 第66-70页 |
| 4.6 The complement of GQ(3,9) | 第70-72页 |
| 4.7 The complement of the McLaughlin graph | 第72-76页 |
| 5 Conclusion, open problems and future work | 第76-80页 |
| 5.1 Contribution | 第76-77页 |
| 5.2 Future work | 第77-78页 |
| 5.3 Open problems | 第78-80页 |
| References | 第80-85页 |
| Research work conducted during PhD | 第85页 |
