molecule theory and quantum chemical topology. Four new approaches, including analytical minimal molecular surface, Hessian matrix eigenvalue map, curvature map and virtual particle model, are introduced for the first time to bridge the gaps in biomolecular modeling and analysis. Emphasis is given to the connections of existing biophysical models/methods to mathematical theories, such as graph theory, Morse theory, Poicaré-Hopf theorem, differential geometry, differential topology, algebraic topology and geometric topology. Potential new directions and standing open problems are briefly discussed.'
author:
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    Kelin Xia$^1$ [^1] and Guo-Wei Wei$^{2,3}$ [^2]\
    $^1$Division of Mathematical Sciences, School of Physical and Mathematical Sciences,\
    Nanyang Technological University, Singapore 637371\
    $^2$Department of Mathematics\
    Michigan State University, MI 48824, USA\
    $^3$Department of Biochemistry & Molecular Biology\
    Michigan State University, MI 48824, USA\
title: 'A review of geometric, topological and graph theory apparatuses for the modeling and analysis of biomolecular data '
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Key words: Molecular bioscience, Molecular biophysics, Graph theory, Graph spectral theory, Graph Laplacian, Differential geometry, Differential topology Persistent homology, Morse theory, Conley index, Poincaré-Hopf index, Laplace-Beltrami operator, Dynamical system.

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Introduction
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Life science is regarded as the last forefront in natural science and the 21$^{\rm st}$ century will be the century of biological 