Background; 1. Introduction; 2. Parter-Wiener, etc. theory; 3. Maximum multiplicity for trees, I; 4. Multiple eigenvalues and structure; 5. Maximum multiplicity, II; 6. The minimum number of distinct eigenvalues; 7. Construction techniques; 8. Multiplicity lists for generalized stars; 9. Double generalized stars; 10. Linear trees; 11. Non-trees; 12. Geometric multiplicities for general matrices over a field.
'The authors offer a unique and modern exploration into the eigenvalues associated with a graph, well beyond the classical treatments. This well-written and comprehensive monograph is ideal for newcomers to this subject and will be beneficial for experienced practitioners as well.' Shaun M. Fallat, University of Regina, Canada 'The undirected graph of a real symmetric matrix tells you the sparsity structure of the matrix. That seems too little information to constrain the eigenvalues. Nevertheless as the matrix gets sparser some constraints appear, not on the actual eigenvalues but on their (algebraic) multiplicities. When the graph is sparse enough to be a tree there is a lot to say. The authors have collected scattered results, filled in key omissions, imposed systematic notation and concepts so that a rich and subtle theory, blending trees and matrices, unfolds before the reader. I, for one, am grateful.' Beresford Parlett, University of California, Berkeley 'This book provides a comprehensive survey and fresh perspectives on a fundamental inverse problem: how does the structure of a matrix impact its spectral properties? The inclusion of recently developed techniques, results and open questions will foster future research and applications.' Bryan Shader, University of Wyoming.