ABSTRACT
Directed graphs, often called digraphs, have much in common with graphs, which were the subject of the previous chapter. While digraphs are of interest in their own right, and have been the subject of much research, this chapter focuses on those aspects of digraphs that are most useful to matrix theory. In particular, it will be seen that digraphs can be used to understand how the zero–nonzero structure of square matrices affects matrix products, determinants, inverses, and eigenstructure. Basic material on digraphs and their adjacency matrices can be found in many texts on graph theory, nonnegative matrix theory, or combinatorial matrix theory. For all aspects of digraphs, except their spectra, see [BG00]. Perhaps the most comprehensive single source for results, proofs, and references to original papers on the interplay between digraphs and matrices is [BR91, Chapters 3 and 9]. Readers preferring a matrix analytic rather than combinatorial approach to irreducibility, primitivity, and their consequences, should consult [BP94, Chapter 2].
