Computational nanotechnology provides theoretical understanding of complex nanoparticles properties and may enlighten their potential applications. In this paper, we use graph algorithms based on the dynamic programming to develop a software system which computes a set of optimal physical properties of the C20 nanoparticle. The computational results of our developed software system have not been reported previously. C20 nanoparticle is the smallest and simplest fullerene composed of twenty equivalent carbon atoms arranged in a cage structural form. The twenty carbon atoms of C20 nanoparticle are also arranged in two other major structures: bowl and ring. The study considers all three major structures of C20 nanoparticle and computes their physical indices including Wiener, hyper-Wiener, Harary and reciprocal Wiener. The Hosoya and hyper-Hosoya polynomials of the cage, bowl and ring structures of C20 nanoparticle have been also computed. The accuracy of the computational results was proved with the mathematical equations between the indices and polynomials. Our computational results also showed good agreement with the physical properties of the cage, bowl and ring structures of C20 nanoparticle.