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Summary
This summary is machine-generated.

This study develops persistent path and hypergraph Dirac operators to analyze topological structures. These novel operators effectively distinguish harmonic spectra and reveal molecular complexity in topological data analysis.

Keywords:
Persistent hypergraph DiracPrimary: 62R40Secondary: 55N31persistent digraph Diracsimultaneous geometric and topological analysesspectral data analysistopological data analysis

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Area of Science:

  • Mathematics
  • Topological Data Analysis
  • Computational Chemistry

Background:

  • Harmonic and non-harmonic spectra analysis is crucial for understanding complex structures.
  • Topological Data Analysis (TDA) offers powerful tools for characterizing data shape and features.
  • Existing methods may lack the sensitivity to capture intricate topological changes.

Purpose of the Study:

  • To develop and explore the persistence of path Dirac and hypergraph Dirac operators.
  • To demonstrate the operators' ability to distinguish harmonic and non-harmonic spectra.
  • To apply these persistent operators to analyze molecular structures within topological data analysis.

Main Methods:

  • Development of path Dirac and hypergraph Dirac operators.
  • Investigation of operator persistence and sensitivity to filtration.
  • Application to graphs and digraphs derived from molecular structures and preorders.

Main Results:

  • The developed operators effectively differentiate between harmonic and non-harmonic spectra.
  • Operator persistence reveals insights into subcomplexes and topological changes.
  • The study demonstrates the utility of persistent operators in analyzing molecular complexity.

Conclusions:

  • Persistent path and hypergraph Dirac operators are effective tools for topological data analysis.
  • These operators provide a nuanced understanding of spectral properties and structural complexity.
  • The application in molecular science highlights their potential for data-driven discovery.