Let G be a simple graph with vertex set V (G) and edge set E(G). The multiplicative forgotten topological index of G are defined as:  F (G) = Y v2V (G) dG(v)3; where dG(v) is the degree of the vertex v of G. In this paper, we present upper bounds for the multiplicative forgotten topological index of several graph operations such as sum, Cartesian product, corona product, composition, strong product, disjunction and symmetric difference in terms of the F􀀀 index and the first Zagreb index of their components. Also, we give explicit formulas for this new graph invariant under two graph operations such as union and Tensor product. Moreover, we obtain the expressions for this new graph invariant of subdivision graphs and vertex – semitotal graphs. Finally, we compare the discriminating ability of indices.

The forgotten topological coindex (also called Lanzhou index) is defined for a simple connected graph G as the sum of the terms du2+dv2 over all non-adjacent vertex pairs uv of G, where du denotes the degree of the vertex u in G. In this paper, we present some inequalities for the forgotten topological coindex in terms of some graph parameters such as the order, size, number of pendent vertices, minimal and maximal vertex degrees, and minimal non-pendent vertex degree. We also study the relation between this invariant and some well-known graph invariants such as the Zagreb indices and coindices, multiplicative Zagreb indices and coindices, Zagreb eccentricity indices, eccentric connectivity index and coindex, and total eccentricity. Exact formulae for computing the forgotten topological coindex of double graphs and extended double cover of a given graph are also proposed.

A topological index is a special number which describes the whole structure of a graph.The topological indices are categorized on the basis of their logical roots from topological invariant. A topological index which depends on the degree of a vertex is called degree based topological index. In this paper we will calculate some degree based multiplicative topological indices of single-walled Titania nanotubes.

The forgotten topological index of a molecular graph G is defined as F (G) = S nÎv (G) d (v)3, where d (v) denotes the degree of vertex v in G. the first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number g = 1, 2, the first through the fourth for g = 3, and the first and the second for g = 4, 5 are determined. These results are compared with those obtained for the first Zagreb index.

For a graph G with edge set E (G), the multiplicative second Zagreb index of G is defined as Õ2(G)=ÕuvÎE(G) [dG (u) dG (v)], where dG (v) is the degree of vertex v in G. In this paper, we identify the eighth class of trees, with the rst through eighth smallest multiplicative second Zagreb indices among all trees of order n³14.

Let G = (E(G),V (G)) be a (molecular) graph with vertex set V (G) and edge set E(G). The forgotten Zagreb index and the hyper Zagreb index of G are defined by F(G) =∑,u, V (G) d(u)3 and HM(G) =∑,uv, E(G)(d(u) + d(v))2 where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. A recent problem called the inverse problem deals with the numerical realizations of topological indices. We see that there exist trees for all even positive integers with F(G) > 88 and with HM(G) > 158. Along with the result, we show that there exist no trees with F(G) < 90 and HM(G) < 160 with some exceptional even positive integers and hence characterize the forgotten Zagreb index and the hyper Zagreb index for trees.

Gutman recently introduced a new vertex-degree-based topological index called the Sombor index. In this paper, we present some new results relating the Sombor index and some well-studied topological indices: Zagreb indices, forgotten index, harmonic index, (general) sum-connectivity index and symmetric division deg index.

‎The second multiplicative Zagreb eccentricity index $E^{*}_{2} ({G})$ of a simple connected‎ graph $G$ is expressed as the product of the weights‎ $\varepsilon_{G}(a)\varepsilon_{G}(b)$ over all edges $ab$ of $G$‎, where $\varepsilon_{G}(a)$ stands for the‎ eccentricity of the vertex $a$ in $G$‎. ‎In this‎ paper‎, ‎some extremal problems on the $E^{*}_{2}$ index over some special graph classes including‎ trees‎, ‎unicyclic graphs and bicyclic graphs are examined‎, ‎and‎ ‎the corresponding extremal graphs are characterized‎. ‎Besides‎, ‎the relationships between this vertex-eccentricity-based graph invariant and some well-known parameters of graphs and existing graph invariants such as the number of vertices‎, ‎number of edges‎, ‎minimum vertex degree‎, ‎maximum vertex degree‎, ‎eccentric connectivity index‎, ‎connective eccentricity index‎, ‎first multiplicative Zagreb eccentricity index and second multiplicative Zagreb index are investigated‎.