Let G and H be graphs. The tensor product G⊗H of G and H has vertex set V (G ⊗ H) = V (G) × V (H) and edge set E(G ⊗ H) = {(a, b)(c, d)|ac ∈ E(G) and bd ∈ E(H)}. In this paper, some results on this product are obtained by which it is possible to compute the Wiener and Hyper Wiener indices of Kn ⊗G.

If G is a connected graph, then the distance between two edges is, by definition, the distance between the corresponding vertices of the line graph of G. The edge-Wiener index We of G is then equal to the sum of distances between all pairs of edges of G. We give bounds on We in terms of order and size. In particular we prove the asymptotically sharp upper bound We(G) ≤ 25 55 n5 + O(n9/2) for gr...

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...

In this paper, we present the exact formulae for the multiplicative version of degree distance and the multiplicative version of Gutman index of strong product of graphs in terms of other graph invariants including the Wiener index and Zagreb index. Finally, we apply our results to the multiplicative version of degree distance and the multiplicative version of Gutman index of open and closed fe...

For a connected graph G and an non-empty set S ⊆ V (G), the Steiner distance dG(S) among the vertices of S is defined as the minimum size among all connected subgraphs whose vertex sets contain S. This concept represents a natural generalization of the concept of classical graph distance. Recently, the Steiner Wiener index of a graph was introduced by replacing the classical graph distance used...

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 degree distance of a graph G is     , 1 1 1 2 n n i j i i j D G d d L        j , where and i d j d are the degrees of vertices , and is the distance between them. The Wiener index is defined as   , i j v v V G  , i j L   , 1 1 1 2 n n i j i j W G L      . An elegant result (Gutman; Klein, Mihalić,, Plavšić and Trinajstić) is known regarding their correlation, that   ...

the wiener polarity index wp(g) of a molecular graph g of order n is the number ofunordered pairs of vertices u, v of g such that the distance d(u,v) between u and v is 3. in anearlier paper, some extremal properties of this graph invariant in the class of catacondensedhexagonal systems and fullerene graphs were investigated. in this paper, some new bounds forthis graph invariant are presented....

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...