Let H be a connected graph with vertex and edge sets V(H) and E(H), respectively. As usual, the distance between the vertices u and v of H is denoted by d(u,v) and it is defined as the number of edges in a minimal path connecting the vertices u and v. A topological index is a real number related to a graph. It must be a structural invariant, i.e., it preserves by every graph automorphisms. Ther...

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.

the evaluation of the edge effects of forest roads on ecosystem is very useful to understand changes from forest road. with this regard, in this research the effect of main forest road construction on regeneration and biodiversity of established trees in edge of forest road was investigated at 35 m distance into forest area. an existing 10 years old road segment, with same physiographic propert...

The Wiener index W (G) of a connected graph G is defined as W (G) = ∑ u,v∈V (G) dG(u, v) where dG(u, v) is the distance between the vertices u and v of G. For S ⊆ V (G), the Steiner distance d(S) of the vertices of S is the minimum size of a connected subgraph of G whose vertex set is S. The k-th Steiner Wiener index SWk(G) of G is defined as SWk(G) = ∑ S⊆V (G) |S|=k d(S). We establish expressi...

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...

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...

Let $G*H$ be the product $*$ of $G$ and $H$. In this paper we determine the rth power of the graph $G*H$ in terms of $G^r, H^r$ and $G^r*H^r$, when $*$ is the join, Cartesian, symmetric difference, disjunctive, composition, skew and corona product. Then we solve the equation $(G*H)^r=G^r*H^r$. We also compute the Wiener index and Wiener polarity index of the skew product.

The sum of distances between all vertices pairs in a connected graph is known as the Wiener Index. It is the earliest of the indices that correlates well with many physicochemical properties of organic compounds and as such has been well-studied over the last quarter of a century. A q-analogue of this index, termed the Wiener Polynomial by Hosoya but also known today as the Hosoya Polynomial, e...

Chemical compounds and drugs are often modeled as graphs where each vertex represents an atom of molecule, and covalent bounds between atoms are represented by edges between the corresponding vertices. This graph derived from a chemical compounds is often called its molecular graph, and can be different structures. In this paper, we determine the logarithm multiplicative Wiener index and recipr...