Algebraic Conditions for Generating Accurate Adjacency Arrays

Data processing systems impose multiple views on data as it is processed by the system. These views include spreadsheets, databases, matrices, and graphs. Associative arrays unify and simplify these different approaches into a common two-dimensional view of data. Graph construction, a fundamental operation in the data processing pipeline, is typically done by multiplying the incidence array representations of a graph, $\mathbf{E}_\mathrm{in}$ and $\mathbf{E}_\mathrm{out}$, to produce an adjacency matrix of the graph that can be processed with a variety of machine learning clustering techniques. This work focuses on establishing the mathematical criteria to ensure that the matrix product $\mathbf{E}_\mathrm{out}^\intercal\mathbf{E}_\mathrm{in}$ is the adjacency array of the graph. It will then be shown that these criteria are also necessary and sufficient for the remaining nonzero product of incidence arrays, $\mathbf{E}_\mathrm{in}^\intercal\mathbf{E}_\mathrm{out}$ to be the adjacency matrices of the reversed graph. Algebraic structures that comply with the criteria will be identified and discussed.