most other flexibility approaches. The above graph theory based methods have been improved in a number of aspects, including crystal periodicity and cofactor corrections [@Kundu:2002; @Kondrashov:2007; @Hinsen:2008; @GSong:2007], and density - cluster rotational - translational blocking [@Demerdash:2012]. These approaches have many applications in biophysics, including stability analysis [@Livesay:2004], molecular docking [@Gerek:2010], and viral capsid analysis [@Rader:2005; @Tama:2005]. In particular, based on spectral graph theory that the behavior of the second eigenmode can be used for clustering, these methods have been utilized to unveil the molecular mechanism of the protein domain motions of hemoglobin [@CXu:2003], F1 ATPase [@WZheng:2003; @QCui:2004], chaperonin GroEL [@Keskin:2002; @WZheng:2007] and the ribosome [@Tama:2003; @YWang:2004]. The reader is referred to reviews for more details [@JMa:2005; @LWYang:2008; @Skjaven:2009; @QCui:2010].

Note that ENM type of methods is still too expensive for analyzing subcellular organelles and multiportein complexes, such as HIV and Zika virus, and molecular motors, due to their matrix decomposition procedure which is of the order of ${\cal O}(N^3)$ in computational complexity, where $N$ is the number of network nodes, or protein atoms. An interesting and important mathematical issue is how to reduce the computational complexity of ENM, GNM and ANM for handling excessively large biomolecules. Flexibility-rigidity index (FRI) 