Matrix operations and the properties of networks and directed graphs

Matrix multiplication is frequently used to derive properties and quantitative features of directed graphs and networks from their adjacency or value matrices. In many cases, a “modified” matrix multiplication is employed, where other binary operations replace ordinary addition and multiplication. In this paper, a set of conditions for permissible value domains, “addition” and “multiplication” operations, and value matrices is developed under which generalized matrix multiplication directly yields properties of a digraph or network. An additional condition ensures that the properties are related to paths of the structure rather than sequences in general. Finally, a number of applications of generalized matrix multiplication are discussed.