Laplacian Eigenvectors of Graphs: Perron-Frobenius and Faber-Krahn Type Theorems (Lecture Notes in Mathematics Book 1915)

Eigenvectors of graph Laplacians have not, to date, been the subject of expository articles and thus they may seem a surprising topic for a book. The authors propose two motivations for this new LNM volume: (1) There are fascinating subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs. (2) "Geometric" properties of (cost) functions defined on the vertex sets of graphs are of practical interest for heuristic optimization algorithms. The observation that the cost functions of quite a few of the well-studied combinatorial optimization problems are eigenvectors of associated graph Laplacians has prompted the investigation of such eigenvectors. The volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology. This Fascinating Volume Investigates The Structure Of Eigenvectors And Looks At The Number Of Their Sign Graphs (nodal Domains), Perron Components, And Graphs With Extremal Properties With Respect To Eigenvectors. The Rayleigh Quotient And Rearrangement Of Graphs Form The Main Methodology. Eigenvectors Of Graph Laplacians May Seem A Surprising Topic For A Book, But The Authors Show That There Are Subtle Differences Between The Properties Of Solutions Of Schrödinger Equations On Manifolds On The One Hand, And Their Discrete Analogs On Graphs. Front Matter....Pages I-VIII Introduction....Pages 1-14 Graph Laplacians....Pages 15-27 Eigenfunctions and Nodal Domains....Pages 29-47 Nodal Domain Theorems for Special Graph Classes....Pages 49-65 Computational Experiments....Pages 67-75 Faber-Krahn Type Inequalities....Pages 77-91 Back Matter....Pages 93-115 Introduction -- Graph Laplacians -- Eigenfunctions And Nodal Domains -- Nodal Domain Theorems For Special Graph Classes -- Computational Experiments -- Faber-krahn Type Inequalities. Türker Bıyıkoğlu, Josef Leydold, Peter F. Stadler. Includes Bibliographical References (p. [101]-114) And Index.