| Preface |
| Invariants / 1： |
| Definitions |
| Pros and cons of pictures |
| Different versus nonisomorphic graphs |
| Isomorphism problem, invariants |
| "first theorem of graph theory" |
| Adjacency matrix |
| Chromatic Number / 2： |
| Proper colorings |
| Fundamental Counting Principle |
| Independence, clique, and chromatic numbers |
| Chromatic polynomial |
| Unions and joins |
| Bipartite graphs, trees |
| Paraffins, Wiener and Balaban (chemical) indices |
| Connectivity / 3： |
| Quantitative measures of connectivity |
| Blocks, k-connectedness |
| Separating sets, internally disjoint paths, Menger's Theorem |
| Whitney's Broken Cycle Theorem |
| Planar Graphs / 4： |
| Euler's Formula |
| Kuratowski's Theorem |
| Five and Four-Color Theorems |
| Geometric dual |
| Embeddings in orientable surfaces, graph genus |
| Theorems of Heawood and of Ringel and Youngs |
| Hamiltonian Cycles / 5： |
| Necessary conditions |
| Sufficient conditions |
| Closure |
| Chvatal graphs |
| Hamiltonian plane graphs |
| Theorems of Whitney and Grinberg |
| Matchings / 6： |
| Kekule and benzene |
| Perfect matchings |
| Matching polynomial |
| Adjacency characteristic polynomial |
| Matching and covering numbers, Egervary--Konig Theorem from Menger's Theorem |
| Theorems of Hall and Tutte |
| Graphic Sequences / 7： |
| Partitions, graphic partitions, Havel--Hakimi criterion |
| Graphs with the same degree sequence, Ryser switches |
| Majorization, Ferrers diagrams |
| Ruch--Gutman criterion |
| Threshold partitions and graphs |
| Chordal Graphs / 8： |
| Weak majorization, shifted shapes, Ruch--Gutman criterion revealed |
| Split partitions and graphs |
| Chordal graphs |
| Perfect graphs |
| Simplicial vertices and chromatic polynomials |
| Oriented Graphs / 9： |
| Acyclic orientations, Stanley's Theorem |
| Laplacian matrix, spanning tree number |
| Spectral characterization of certain graph invariants |
| Isospectral and decomposable graphs |
| Edge Colorings / 10： |
| Ramsey numbers, upper and lower bounds, Erdos's probabilistic technique |
| Polya-Redfield approach to graph enumeration, cycle index polynomial |
| Hints and Answers to Selected Odd-Numbered Exercises |
| Bibliography |
| Index |
| Index of Notation |
| Preface |
| Invariants / 1： |
| Definitions |
| Pros and cons of pictures |
| Different versus nonisomorphic graphs |
| Isomorphism problem, invariants |
