The theory of graphs, with its diverse applications in natural (Chemistry, Biology) and social sciences in general and in theoretical computer science in particular, is becoming an important component of the mathematics curriculum in colleges and universities all over the world. In this paper we present some theoretical results about some topological indices, like the Wiener index W, Degree distance index DD and the Hyper-Wiener index WW of a graph G. In the application section we are going to apply these theoretic results, for the two first levels of the Spider's Web planar graph R _n and the two first levels of the Network planar graph G _n , to give their Wiener index, Degree distance index and Hyper-Wiener index.