  This work introduces the development of path Dirac and hypergraph Dirac
operators, along with an exploration of their persistence. These operators
excel in distinguishing between harmonic and non-harmonic spectra, offering
valuable insights into the subcomplexes within these structures. The paper
showcases the functionality of these operators through a series of examples in
various contexts. An important facet of this research involves examining the
operators' sensitivity to filtration, emphasizing their capacity to adapt to
topological changes. The paper also explores a significant application of
persistent path Dirac and persistent hypergraph Dirac in the field of molecular
science, specifically in the analysis of molecular structures. The study
introduces strict preorders derived from molecular structures, which generate
graphs and digraphs with intricate path structures. The depth of information
within these path complexes reflects the complexity of different preorder
classes influenced by molecular structures. This characteristic underscores the
effectiveness of these tools in the realm of topological data analysis.
