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This work introduces the development of path Dirac and hypergraph Diracoperators, along with an exploration of their persistence. These operatorsexcel in distinguishing between harmonic and non-harmonic spectra, offeringvaluable insights into the subcomplexes within these structures. The papershowcases the functionality of these operators through a series of examples invarious contexts. An important facet of this research involves examining theoperators' sensitivity to filtration, emphasizing their capacity to adapt totopological changes. The paper also explores a significant application ofpersistent path Dirac and persistent hypergraph Dirac in the field of molecularscience, specifically in the analysis of molecular structures. The studyintroduces strict preorders derived from molecular structures, which generategraphs and digraphs with intricate path structures. The depth of informationwithin these path complexes reflects the complexity of different preorderclasses influenced by molecular structures. This characteristic underscores theeffectiveness of these tools in the realm of topological data analysis.

Persistent Dirac of Path and Hypergraph