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Topological index is a numerical descriptor of a molecule, based on a certain topological feature of the corresponding molecular graph, it is found that there is a strong correlation between the properties of chemical compounds and their molecular structure. In the other side, titania nanotube is a well-known semiconductor and has numerous technological applications such as biomedical devices, dye-sensitized solar cells, and etc. In this paper, the second Hyper-Zagreb index and their coindex of Titania nanotubes have been computed. Furthermore, strong correlation coefficients between second Hyper-Zagreb index and some physicochemical properties such as Density (DENS), Molar volume (MV), Acentric factor (Acenfac) and Entropy (S) have been Appeared.

A chemical graph theory is a fascinating branch of graph theory which has many applications related to chemistry. A topological index is a real number related to a graph, as its considered a structural invariant. It’s found that there is a strong correlation between the properties of chemical compounds and their topological indices. In this paper, we introduce some new graph operations for the first Zagreb index, second Zagreb index and forgotten index "F-index". Furthermore, it was found some possible applications on some new graph operations such as roperties of molecular graphs that resulted by alkanes or cyclic alkanes.

A topological index of graph \(G\) is a numerical parameter related to graph which characterizes its molecular topology and is usually graph invariant. Topological indices are widely used to determine the correlation between the specific properties of molecules and the biological activity with their configuration in the study of quantitative structure-activity relationships (QSARs). In this paper some basic mathematical operations for the forgotten index of complement graph operations such as join \(\overline {G_1+G_2}\), tensor product \(\overline {G_1 \otimes G_2}\), Cartesian product \(\overline {G_1\times G_2}\), composition \(\overline {G_1\circ G_2}\), strong product \(\overline {G_1\ast G_2}\), disjunction \(\overline {G_1\vee G_2}\) and symmetric difference \(\overline {G_1\oplus G_2}\) will be explained. The results are applied to molecular graph of nanotorus and titania nanotubes.

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In this paper, we introduce new binary operations on graphs. In fact, we obtained some other product operations, called them classic product operations from union of two or more new product operations. We examined the relationship between new binary product operations and classic product operations.