‎the harmonic index of a connected graph $g$‎, ‎denoted by $h(g)$‎, ‎is‎ ‎defined as $h(g)=sum_{uvin e(g)}frac{2}{d_u+d_v}$‎ ‎where $d_v$ is the degree of a vertex $v$ in g‎. ‎in this paper‎, ‎expressions for the harary indices of the‎ ‎join‎, ‎corona product‎, ‎cartesian product‎, ‎composition and symmetric difference of graphs are‎ ‎derived‎.

The forgotten topological index is defined as sum of third power of degrees. In this paper, we compute some properties of forgotten index and then we determine it for some classes of product graphs.

A new topological index J (based on distance sums s i as graph invariants) is proposed. For unsaturated or aromatic compounds, fractional bond orders are used in calculating s i. The degeneracy of J is lowest among all single topological indices described so far. The asymptotic behaviour of J is discussed, e.g. when n ~ ~ in CnH2n+2, J ~ ~r for linear alkanes, and J ~ ** for highly branched ones.

the topological indices, defined as the sum of contributions of all pairs of vertices (among which are the wiener, harary, hyper–wiener indices, degree distance, and many others), are expressed in terms of contributions of edges and pairs of edges.

in this paper, we first collect the earlier results about some graph operations and then wepresent applications of these results in working with chemical graphs.

The atom bond connectivity index of a graph is a new topological index was defined by E. Estrada as ABC(G)  uvE (dG(u) dG(v) 2) / dG(u)dG(v) , where G d ( u ) denotes degree of vertex u. In this paper we present some bounds of this new topological index.

the padmakar-ivan (pi) index is a first-generation topological index (ti) based on sums overall edges between numbers of edges closer to one endpoint and numbers of edges closer to theother endpoint. edges at equal distances from the two endpoints are ignored. an analogousdefinition is valid for the wiener index w, with the difference that sums are replaced byproducts. a few other tis are discu...

Let G be a connected simple (molecular) graph. The distance d(u, v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. In this paper we compute some distance based topological indices of H-Phenylenic nanotorus. At first we obtain an exact formula for the Wiener index. As application we calculate the Schultz index and modified Schultz index of this...

Dendrimers are highly branched organic macromolecules with successive layers of branch units surrounding a central core. The M-polynomial of nanotubes has been vastly investigated as it produces many degree-based topological indices. These indices are invariants of the topology of graphs associated with molecular structure of nanomaterials to correlate certain physicochemical properties like bo...

The Wiener index of a graph G is defined as W(G) = 1/2∑{x,y}⊆V(G)d(x,y), where V(G) is the set of all vertices of G and for x,y ∈ V(G), d(x,y) denotes the length of a minimal path between x and y. In this paper, we first report our recent results on computing Wiener, PI and Balaban indices of some nanotubes and nanotori. Next, the PI and Szeged indices of a new type of nanostar dendrimers are c...