In this thesis, we present a new perspective on tight binding models. Utilising the rich algebraic toolkit provided by a combination of graph and matrix theory allows us to explore tight binding systems related through polynomial relationships.

By utilising ring operations of weighted digraphs through intermediate König digraph representations, we establish a polynomial algebra over finite and infinite periodic graphs, analogous to polynomial operations on adjacency matrices. Exploring the microscopic and macroscopic behaviour of polynomials in a graph-theoretic setting, we reveal elegant relationships between the symmetrical, topological, and spectral properties of a parent graph G and its family of child graphs p(G).

Drawing a correspondence between graphs and tight binding models, we investigate deep-rooted connections between different quantum systems, providing a fresh angle from which to view established tight binding models.

Finally, we visit topological chains, demonstrate how their properties relate to more trivial underlying chains through effective “square root” operations, and provide new insights into their spectral characteristics.