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.

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.

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.

‎In this study‎, ‎we find some bounds for the Sombor index of a graph‎ ‎$G$‎ ‎by some topological and statistical indices such as {\it arithmetic index}‎, ‎{\it geometric index}‎, ‎‎ ‎{\it arithmetic-geometric (AG) index}‎, ‎{\it geometric-arithmetic (GA) index}‎, ‎{\it symmetric division degree index (SDD(G))}‎, ‎and some central and dispersion indices‎. ‎The bounds can state estimated values and error intervals for the Somber index and show limits of accuracy‎. ‎Error intervals are expressed as inequalities‎.

Let G be a graph with vertex set V (G) and edge set E(G), and let du denote the degree of vertex u 2 V (G). The Randi c index of G is de ned as R(G) = P 1= p dudv: In this paper, we investigate the relationships between Randi c uv2E(G) index and several topological indices.

In [13] Gutman introduced a novel graph invariant called Sombor index SO, defined via $\sqrt{\deg(u)^{2}+\deg(v)^{2}}.$ In this paper we provide relations between Sombor index and some degree-based topological indices: Zagreb indices, Forgotten index and Randi\' {c} index. Similar relations are established in the class of triangle-free graphs.

Assume denotes a connected and simple graph with edge set  E(G) as well as vertex set ‎V(G). ‎In chemical graph theory‎, ‎the atom-bond connectivity index, as well as the Randić index of graph are two well-defined topological indices‎. ‎In addition‎, ‎Ali and Du [On the difference between  ABC and Randić indices of binary and chemical trees‎, ‎Int‎. ‎J‎. ‎Quantum Chem‎. ‎(2017) e25446] recently unveiled the distinction between Randić and ABC indices‎. ‎In this report‎, ‎we study the link between the difference of Randić and ABC indices with certain well-studied topological indices‎.