Using Adjacency Matrices to Represent Rational Generating Functions

This paper shows that adjacency matrices can be used to represent rational generating functions (RGFs). It is known that a closed form solution exists for computing coefficients of RGFs. Also, one can write the linear recurrence relation associated with every RGF into a matrix format. What has not yet been shown (or is not yet commonly discussed) is that one can conceptualize an RGF as a system of connected cycles within an overarching adjacency matrix. Each cycle corresponds to a factor of the RGF. There may be a benefit to taking the cyclical perspective. For example, certain linear recurrence matrices have cells containing positive and negative values whereas the cyclical approach has cells containing only positive values. The computational benefit is probably irrelevant for computers; however, it may be important for restrictive systems, such as biological systems / neural networks that have a tight operating envelope. We make a final observation that each matrix can be thought of as a graph which is an epsilon away from being strongly connected. In essence, the study of sequences modeled by RGFs can be converted to the study of connected cyclical graphs.