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Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs

Table of Contents

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

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