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Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs

Jason J. Molitierno
Champman & Hall/CRC
Publication Date: 
Number of Pages: 
Discrete Mathematics and Its Applications
[Reviewed by
John T. Saccoman
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This book is part of the series “Discrete Mathematics and its Applications.” It continues the recent line of books that exploit the connections between the two seemingly disparate subjects of graph theory and matrix theory. While some of these books are more along the lines of graduate-level research monographs (such as An Introduction to the Theory of Graph Spectra by Cvetković, Rowlinson, and Simić), or an undergraduate textbook (Graphs and Matrices by Bapat) , 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).

Two other texts, one by Brualdi and Ryser from 1991 (Combinatorial Matrix Theory) and one by Brualdi and Cvetković from 2009 (A Combinatorial Approach to Matrix Theory and Its Applications) have similar titles, but are at a higher level. In the former, such topics as permanents and Latin Squares are given treatment, while the latter discusses canonical forms and applications to electrical engineering, chemistry and physics.

After two chapters covering the preliminaries in Matrix Theory and Graph Theory necessary for the sequel, Molitierno presents an Introduction to Laplacian Matrices, with a proof of the Kirchhoff Matrix-Tree Theorem via Cauchy-Binet. He discusses Laplacians of weighted graphs as well as unweighted ones, and bounds on the eigenvalue spectra of certain classes of graphs. In particular, Molitierno focuses on the second smallest eigenvalue of a graph’s Laplacian matrix, called the algebraic connectivity of the graph.

The important work of Grone and Merris is given a decent treatment, as is Fielder’s. In fact, it is Fiedler’s theorem on eigenvectors that leads to a particular type of matrix that dominates the last two chapters of the book, the so-called “bottleneck matrices.” These matrices are used to determine such graph properties as algebraic connectivity. Chapter 6 covers the bottleneck matrices for trees, while some general classes of non-tree graphs are covered in chapter 7.

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 is Professor of Mathematics at Seton Hall University in South Orange, NJ.

Matrix Theory Preliminaries
Vector Norms, Matrix Norms, and the Spectral Radius of a Matrix
Location of Eigenvalues
Perron-Frobenius Theory
Doubly Stochastic Matrices
Generalized Inverses

Graph Theory Preliminaries
Introduction to Graphs
Operations of Graphs and Special Classes of Graphs
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