Eigenvalues, Multiplicities and Graphs (Cambridge Tracts in Mathematics)
Cambridge University Press, 2/12/2018
EAN 9781107095458, ISBN10: 110709545X
Hardcover, 310 pages, 23.6 x 15.8 x 2.3 cm
Language: English
The arrangement of nonzero entries of a matrix, described by the graph of the matrix, limits the possible geometric multiplicities of the eigenvalues, which are far more limited by this information than algebraic multiplicities or the numerical values of the eigenvalues. This book gives a unified development of how the graph of a symmetric matrix influences the possible multiplicities of its eigenvalues. While the theory is richest in cases where the graph is a tree, work on eigenvalues, multiplicities and graphs has provided the opportunity to identify which ideas have analogs for non-trees, and those for which trees are essential. It gathers and organizes the fundamental ideas to allow students and researchers to easily access and investigate the many interesting questions in the subject.
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.