1st Edition

# Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs

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On the surface, matrix theory and graph theory seem like very different branches of mathematics. However, adjacency, Laplacian, and incidence matrices are commonly used to represent graphs, and many properties of matrices can give us useful information about the structure of graphs.

**Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs** is a compilation of many of the exciting results concerning Laplacian matrices developed since the mid 1970s by well-known mathematicians such as Fallat, Fiedler, Grone, Kirkland, Merris, Mohar, Neumann, Shader, Sunder, and more. The text is complemented by many examples and detailed calculations, and sections followed by exercises to aid the reader in gaining a deeper understanding of the material. Although some exercises are routine, others require a more in-depth analysis of the theorems and ask the reader to prove those that go beyond what was presented in the section.

Matrix-graph theory is a fascinating subject that ties together two seemingly unrelated branches of mathematics. Because it makes use of both the combinatorial properties and the numerical properties of a matrix, this area of mathematics is fertile ground for research at the undergraduate, graduate, and professional levels. This book can serve as exploratory literature for the undergraduate student who is just learning how to do mathematical research, a useful "start-up" book for the graduate student beginning research in matrix-graph theory, and a convenient reference for the more experienced researcher.

**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

### Biography

Jason J. Molitierno

… this book works well as a reference textbook for undergraduates. Indeed, it is a distillation of a number of key results involving, specifically, the Laplacian matrix associated with a graph (which is sometimes called the ‘nodal admittance matrix’ by electrical engineers). … Molitierno’s book represents a well-written source of background on this growing field. The sources are some of the seminal ones in the field, and the book is accessible to undergraduates.

—John T. Saccoman,MAA Reviews, October 2012The book owes its textbook appeal to detailed proofs, a large number of fully elaborated examples and observations, and a handful of exercises, making beginning graduate students as well as advanced undergraduates its primary audience. Still, it can serve as useful reference book for experienced researchers as well.

—Zentralblatt MATH