Let G be a connected undirected graph. A topological index of G is a numerical descriptor of G which is invariant under isomorphism of graphs. Usually topological indices describe structural properties of molecular graphs. In this thesis we compute some of the famous topological indices of TUC 4 C 8 ( R) and TUC 4 C 8 ( S) nanotubes. We compute the Wiener, Hyper Wiener, Schultz and Szeged indices of those nanotubes. We stady rooted product of graphs where induced by Godsil and Mckay in 1978. Let G={G 1 ,G 2 ,…,G n } be a sequence of rooted graphs and H be a labelled graph on n vertices. The rooted product H(G) of H y G, obtained by identifying the root vertices of G i for i=1,2,…,n. We compute Wiener index and characteristic polynomial of adjacency and Laplacian matrices of H(G). As application of our method we compute Wiener index and characteristic polynomial of adjacency and Laplacian matrices of some graphs which represented in term of rooted roduct of simple graphs. Afterward we compute the characteristic polynomial of adjacency matrix of line graph of rooted trees. Also we compute Wiener index and the characteristic polynomial of adjacency and Laplacian matrices of line graph of rooted tree and generalized bethe etres.