Combinatorial Matrix Theory

The 2018 book entitled Combinatorial Matrix Theory, written by Richard Brualdi, Angeles Carmona, Pauline van den Driessche, Stephen Kirkland and Dragan Stevanovic, is not a reissue of the 1991 text that Brualdi wrote with Ryser. It is, however, a very excellent treatment of the subject for mathematicians interested in the intersection between matrix theory and combinatorics. The first Brualdi text bearing this title was, in fact, a textbook, with nine chapters, and exercises at the end of them. The new text is a survey of the current results in the field. As explained in the foreword, the chapters are organized around five series of lectures, each containing four individual lectures, delivered by the coauthors as an advanced course in combinatorial matrix theory at the Centre de Recerca Matematica in Barcelona , Spain from June 29-July 3, 2015.

 

Brualdi himself wrote the first chapter, “Some Combinatorially Defined Matrix Classes,” which deals with Permutation matrices, Alternating Sign matrices and Tournament matrices. Subsequent chapters address “Sign Pattern Matrices” (by van den Dreissche), “Spectral Radius of Graphs” (Stevanovic), “Group Inverse of Laplacian Matrices” (Kirkland) and “Boundary Value Problems on Finite Networks” (Carmona). This new text is somewhat less theoretical than the older one, but as one can glean from the previous paragraph, much more applications-driven. There are numerous figures, and in color, which enhances the reader’s visualization of the concept. 

 

While proofs of some of the results are included, it is the nature of a survey to present as many important results and concepts as possible, sacrificing breadth over depth, if you will. Each chapter, however, contains a lengthy bibliography, which offers avenues for further study, or clarification.  As such, while not a textbook, like its predecessor, it is appropriate for graduate students and others interested in the latest developments in this rich and diverse field.

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John T. Saccoman is Professor in the Department of Mathematics and Computer Science at Seton Hall University in New Jersey. His research interests include graph theory, network reliability theory and sabermetrics.