where m is the number of edges, μ is the cyclomatic number of G, d(u) is thesum of distances between vertex u and all of the vertices of G, and the summationgoes over all edges from the edge set E(G). In this paper we obtain amethod for calculating the Balaban index of nanotubes which have square andoctagon structure and denoted by C4C8(S) nanotubes. Manuscript profile

The Wiener index of a graph G is defined as W(G) = ... where V (G) is the setof all vertices of G and for i,j in V (G), d(i,j) is the minimum distance between i and j. Ashrafiand yousefi (see A. R. Ashrafi and S. Yousefi, Computing the Wiener Index of a TUC4C8(S)Nanotorus, MATCH Commun. Math. Comput. Chem., 57(2)(2007), 403-410) computed theWiener index of TUC4C8(S) Nanotorus. In this paper we use a new method to compute theWiener index of these Nanotorus. Manuscript profile

A generalized Bethe tree is a rooted unweighted tree in whichthe vertices in each of its levels have equal degree. In this paperwe derive an explicit formula for the characteristic polynomialsof the adjacency and Laplacian matrices of unweighted rootedtree which obtained from the union of the generalized Bethe treesjoined at their respective root vertices by using of rooted productof graphs. Manuscript profile