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Combinatorial matrix theory. (English) Zbl 0746.05002
Encyclopedia of Mathematics and Its Applications. 39. Cambridge etc.: Cambridge University Press. ix, 367 p. (1991).
This is the first book length exposition of basic results of matrix theory having combinatorial character. It consists of nine chapters: Incidence matrices, Matrices and graphs, Matrices and digraphs, Matrices and bipartite graphs, Some special graphs, Existence theorems, The permanent, Latin squares, Combinational matrix algebra.
The first five chapters deal with the many connections of graphs, digraphs and matrices. Chapter 6 contains existence theorems for matrices with prescribed combinatorial properties and various matrix decomposition theorems. Chapters 7 and 8 treat the permanent of the square matrix and Latin squares. Chapter 8 contains, in particular, Smetaniuk’s proof of the conjecture of Evans on a possibility to complete a partial Latin square of order \(n\) containing at most \(n-1\) specific elements.
The final chapter deals with algebraic characterization of combinatorial properties and the use of combinatorial arguments in proving such classical theorems as the Cayley-Hamilton theorem and the Jordan canonical form.
The book includes many exercises, making it suitable for use as a graduate text.

05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05B15 Orthogonal arrays, Latin squares, Room squares