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Linear algebra is an important component in the study of graphs. This book illustrates the elegance and power of matrix techniques in this study by means of several results, classical and recent. The emphasis on matrix techniques is greater than in other standard references on algebraic graph theory, and the important matrices associated with graphs such as incidence, adjacency and Laplacian matrices are treated in detail.
Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree and its generalized version for arbitrary graphs, known as resistance matrix. Later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph.
Such an extensive coverage of the subject provides a prompt for further exploration, and the inclusion of exercises throughout the book enables practical learning. This can be applied to a selection of sub-disciplines within science and engineering.
This book will be invaluable to students and researchers in graph theory and combinatorial matrix theory who want to be acquainted with matrix theoretic ideas used in graph theory. It will also benefit a wider, cross-disciplinary readership.
Contents:
1. Preliminaries 2. Incidence Matrix 3. Adjacency Matrix 4. Laplacian Matrix 5. Cycles and Cuts 6. Regular Graphs 7. Algebraic Connectivity 8. Distance Matrix of a Tree 9. Resistance Distance 10. Laplacian Eigenvalues of Threshold Graphs 11. Positive Definite Completion Problem 12. Matrix Games Based on Graphs
Hints and Solutions to Selected Exercises
ISBN - 9789380250076
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