Abstract. A subgraph H of a graph G is gated if for every x ∈ V (G) there exists a vertex u in H such that dG(x, v) = dG(x, u) + dG(u, v) for any v ∈ V (H). The gated amalgam of graphs G1 and G2 is obtained from G1 and G2 by identifying their isomorphic gated subgraphs H1 and H2. Two theorems on the Wiener index of gated amalgams are proved. Several known results on the Wiener index of (chemica...

Let G be a simple graph with vertex set and edge set . The function which assigns to each pair of vertices in , the length of minimal path from to , is called the distance function between two vertices. The distance function between and edge and a vertex is where for and. , . The Wiener index of a graph is denoted by and is defined by .In general this kind of index is called a topological index...

The Wiener index of a connected graph G, denoted by W (G), is defined as 12 ∑ u,v∈V (G) dG(u, v). Similarly, the hyper-Wiener index of a connected graph G, denoted by WW (G), is defined as 1 2W (G) + 1 4 ∑ u,v∈V (G) dG(u, v). The vertex Padmakar-Ivan (vertex PI) index of a graph G is the sum over all edges uv of G of the number of vertices which are not equidistant from u and v. In this paper, ...

Let d(G, k) be the number of pairs of vertices of a graph G that are at distance k, λ a real number, and Wλ(G) = ∑ k≥1 d(G, k)kλ. Wλ(G) is called the Wiener-type invariant of G associated to real number λ. In this paper, the Wiener-type invariants of some graph operations are computed. As immediate consequences, the formulae for reciprocal Wiener index, Harary index, hyperWiener index and Tratc...

The Wiener index of a graph is the sum of the distances between all pairs of vertices. In fact, many mathematicians have study the property of the sum of the distances for many years. Then later, we found that these problems have a pivotal position in studying physical properties and chemical properties of chemical molecules and many other fields. Fruitful results have been achieved on the Wien...

The Wiener index has been studied for simply generated random trees, non-plane unlabeled random trees and a huge subclass of random grid trees containing random binary search trees, random medianof-(2k+ 1) search trees, random m-ary search trees, random quadtrees, random simplex trees, etc. An important class of random grid trees for which the Wiener index was not studied so far are random digi...

We resolve two conjectures of Hri\v{n}\'{a}kov\'{a}, Knor and \v{S}krekovski (2019) concerning the relationship between variable Wiener index Szeged for a connected, non-complete graph, one which would imply other. The strong conjecture is that any such graph there critical exponent in $(0,1]$, below larger above larger. weak always exceeding $1$. They proved bipartite graphs, trees. In this no...

A new extension of the generalized topological indices (GTI) approach is carried out to represent “simple” and “composite” topological indices (TIs) in an unified way. This approach defines a GTI-space from which both simple and composite TIs represent particular subspaces. Accordingly, simple TIs such as Wiener, Balaban, Zagreb, Harary and Randić connectivity indices are expressed by means of ...

The Wiener index, denoted byW (G), of a connected graph G is the sum of all pairwise distances of vertices of the graph, that is, W (G) = 1 2 ∑ u,v∈V (G) d(u, v). In this paper, we obtain the Wiener index of the tensor product of a path and a cycle.