Paths in directed graphs and spectral properties of matrices

AbstractThe study of the relationship between graph theoretic properties and spectral properties of matrices has been of interest in the past eighty years. For about seventy years, research focused on nonnegative matrices, but in the past decade the investigation has been extended to general matrices over an arbitrary field. Several papers have been devoted to the relationship between the structure of the Jordan blocks associated with the eigenvalue 0 of a matrix and the digraph of the matrix. This problem appears to be strongly linked to a graph theoretic study of paths in digraphs. This paper reviews the development of the above- mentioned studies. It proceeds simultaneously with the graph theoretic and the matrix theoretic problems, showing the links at each stage