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Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs
Jason J. Molitierno
Publisher: Champman & Hall/CRC (2012)
Details: 405 pages, Hardcover
Series: Discrete Mathematics and Its Applications
Topics: Graph Theory, Linear Algebra
MAA Review[Reviewed by John T. Saccoman, on 10/02/2012]
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.