Eigenvalues, Multiplicities and Graphs

This book investigates the influence of the graph of a symmetric matrix on the multiplicities of its eigenvalues.

Charles R. Johnson (Author), Carlos M. Saiago (Author)

9781107095458, Cambridge University Press

Hardback, published 12 February 2018

310 pages
23.6 x 15.8 x 2.3 cm, 0.56 kg

'In this book, the multiplicity of eigenvalues in graph theory is discussed in detail. No matter the theoretical knowledge or the way of thinking of the reader, this book has good learning and reference value. At the same time, it also plays a great role in readers' study and research. Therefore, it is a meaningful book.' Xiaogang Liu, zbMATH

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.

Subject Areas: Maths for scientists [PDE], Science: general issues [PD], Number theory [PBH], Mathematical foundations [PBC], Mathematics [PB]