Diamond condition for commuting adjacency matrices of directed and undirected graphs

In the context of the stability analysis of interdependent networks through the eigenvalue evaluation of their adjacency matrices, we characterize algebraically and also geometrically necessary and sufficient conditions for the adjacency matrices of directed and undirected graphs to commute. We also discuss the problem of communicating the concepts, the theorems, and the results to a non-mathematical audience, and more generally across different disciplinary domains, as one of the fundamental challenges faced by the Internet Science community. Thus, the paper provides much more background, discussion, and detail than would normally be found in a purely mathematical publication, for which the proof of the diamond condition would require only a few lines. Graphical visualization, examples, discussion of important steps in the proof and of the diamond condition itself as it applies to graphs whose adjacency matrices commute are provided. The paper also discusses interdependent graphs and applies the results on commuting adjacency matrices to study when the interconnection matrix encoding links between two disjoint graphs commutes with the adjacency matrix of the disjoint union of the two graphs. Expected applications are in the design and analysis of interdependent networks