''Preface On the surface, matrix theory and graph theory are seemingly very different branches of mathematics. However, these two branches of mathematics interact since it is often convenient to represent a graph as a matrix. Adjacency, Laplacian, and incidence matrices are commonly used to represent graphs. In 1973, Fiedler published his first paper on Laplacian matrices of graphs and showed how many properties of the Laplacian matrix, especially the eigenvalues, can give us useful information about the structure of the graph. Since then, many papers have been published on Laplacian matrices. This book is a compilation of many of the exciting results concerning Laplacian matrices that have been developed since the mid 1970's. Papers written by well-known mathematicians such as (alphabetically) Fallat, Fiedler, Grone, Kirkland, Merris, Mohar, Neumann, Shader, Sunder, and several others are consolidated here. Each theorem is referenced to its appropriate paper so that the reader can easily do more in-depth research on any topic of interest. However, the style of presentation in this book is not meant to be that of a journal but rather a reference textbook. Therefore, more examples and more detailed calculations are presented in this book than would be in a journal article. Additionally, most sections are followed by exercises to aid the reader in gaining a deeper understanding of the material. Some exercises are routine calculations that involve applying the theorems presented in the section. Other exercises require a more in-depth analysis of the theorems and require the reader to prove theorems that go beyond what was presented in the section. Many of these exercises are taken from relevant papers and they are referenced accordingly''-- Read more...
Content: Matrix Theory Preliminaries Vector Norms, Matrix Norms, and the Spectral Radius of a Matrix Location of Eigenvalues Perron-Frobenius Theory M-Matrices Doubly Stochastic Matrices Generalized Inverses Graph Theory Preliminaries Introduction to Graphs Operations of Graphs and Special Classes of Graphs Trees Connectivity of Graphs Degree Sequences and Maximal Graphs Planar Graphs and Graphs of Higher Genus Introduction to Laplacian Matrices Matrix Representations of Graphs The Matrix Tree Theorem The Continuous Version of the Laplacian Graph Representations and Energy Laplacian Matrices and Networks The Spectra of Laplacian Matrices The Spectra of Laplacian Matrices Under Certain Graph Operations Upper Bounds on the Set of Laplacian Eigenvalues The Distribution of Eigenvalues Less than One and Greater than One The Grone-Merris Conjecture Maximal (Threshold) Graphs and Integer Spectra Graphs with Distinct Integer Spectra The Algebraic Connectivity Introduction to the Algebraic Connectivity of Graphs The Algebraic Connectivity as a Function of Edge Weight The Algebraic Connectivity with Regard to Distances and Diameters The Algebraic Connectivity in Terms of Edge Density and the Isoperimetric Number The Algebraic Connectivity of Planar Graphs The Algebraic Connectivity as a Function Genus k where k is greater than 1 The Fiedler Vector and Bottleneck Matrices for Trees The Characteristic Valuation of Vertices Bottleneck Matrices for Trees Excursion: Nonisomorphic Branches in Type I Trees Perturbation Results Applied to Extremizing the Algebraic Connectivity of Trees Application: Joining Two Trees by an Edge of Infinite Weight The Characteristic Elements of a Tree The Spectral Radius of Submatrices of Laplacian Matrices for Trees Bottleneck Matrices for Graphs Constructing Bottleneck Matrices for Graphs Perron Components of Graphs Minimizing the Algebraic Connectivity of Graphs with Fixed Girth Maximizing the Algebraic Connectivity of Unicyclic Graphs with Fixed Girth Application: The Algebraic Connectivity and the Number of Cut Vertices The Spectral Radius of Submatrices of Laplacian Matrices for Graphs The Group Inverse of the Laplacian Matrix Constructing the Group Inverse for a Laplacian Matrix of a Weighted Tree The Zenger Function as a Lower Bound on the Algebraic Connectivity The Case of the Zenger Equalling the Algebraic Connectivity in Trees Application: The Second Derivative of the Algebraic Connectivity as a Function of Edge Weight
Abstract: ''Preface On the surface, matrix theory and graph theory are seemingly very different branches of mathematics. However, these two branches of mathematics interact since it is often convenient to represent a graph as a matrix. Adjacency, Laplacian, and incidence matrices are commonly used to represent graphs. In 1973, Fiedler published his first paper on Laplacian matrices of graphs and showed how many properties of the Laplacian matrix, especially the eigenvalues, can give us useful information about the structure of the graph. Since then, many papers have been published on Laplacian matrices. This book is a compilation of many of the exciting results concerning Laplacian matrices that have been developed since the mid 1970's. Papers written by well-known mathematicians such as (alphabetically) Fallat, Fiedler, Grone, Kirkland, Merris, Mohar, Neumann, Shader, Sunder, and several others are consolidated here. Each theorem is referenced to its appropriate paper so that the reader can easily do more in-depth research on any topic of interest. However, the style of presentation in this book is not meant to be that of a journal but rather a reference textbook. Therefore, more examples and more detailed calculations are presented in this book than would be in a journal article. Additionally, most sections are followed by exercises to aid the reader in gaining a deeper understanding of the material. Some exercises are routine calculations that involve applying the theorems presented in the section. Other exercises require a more in-depth analysis of the theorems and require the reader to prove theorems that go beyond what was presented in the section. Many of these exercises are taken from relevant papers and they are referenced accordingly''
