apparatuses, namely, graph theory and persistent homology, for the analysis and modeling of biomolecular data. More specifically, we illustrate the applications of graph theory, Gaussian network model, anisotropic network model, normal mode analysis, flexibility and rigidity index, spectral graph theory, and persistent homology to biomolecular data analysis, such as protein B-factor prediction, domain separation, anisotropic motion, topological fingerprints, etc. The review of some continuous geometric and topological apparatuses for scalar field topology and geometry are given in Section \[sec:continuous\]. We examine the basic concepts in differential geometry, biomolecular surfaces, curvature analysis, and theory of atoms in molecules. Discrete to continuum mapping and two algorithms for curvature evaluation are discussed. Further, we introduce two new approaches, i.e., Hessian matrix eigenvalue maps and curvature maps, for geometric-topological fingerprint analysis of biomolecular data. The relation between scalar field geometry and topological CPs are discussed in detail. A new analytical minimal molecular surface is also introduced. Virtual particle model is proposed to analyze the anisotropic motions of continuous scalar fields, such as cryo-EM maps. Finally, persistent homology analysis for eigenvalue scalar field is discussed. This paper ends with some concluding remarks.

Discrete apparatuses for biomolecules {#sec:discrete}
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One of the major challenges in the 