Given a simple connected undirected graph G, the Wiener index W (G) of G is defined as half the sum of the distances over all pairs of vertices of G. In practice, G corresponds to what is known as the molecular graph of an organic compound. We obtain a sharp lower bound for W (G) of an arbitrary graph in terms of the order, size and diameter of G.

Graphical matrices are presented. Their construction via selected sets of subgraphs and the replacement of subgraphs by numbers representing graph invariants are discussed. The last step of the procedure is to apply the method of choice for obtaining the desired double invariant from the graphical matrix in the numerical form. It is also pointed out that many so-called special graph-theoretical...

topological indices are one of the oldest and most widely used descriptors in quantitative structureproperties relationvhips (qspr). amongst the topological indices used a,s descriptors in qspic., the wienerindex is by far the most popular index. as it has been shown that the wiener index has a strong correlationwith the chemical propenies of the compound.in this study, the relationship between...

The Wiener index of a graph is the sum of all pairwise distances of vertices of the graph. In this paper we characterize the trees which minimize the Wiener index among all trees of given order and maximum degree and the trees which maximize the Wiener index among all trees of given order that have only vertices of two di erent degrees.

The Wiener index of a connected graph is the sum of topological distances between all pairs of vertices. Since Wang in [23] gave a mistake result on the maximum Wiener index for given tree degree sequence, in this paper, we investigate the maximum Wiener index of trees with given degree sequences and extremal trees which attain the maximum value.

Let (G,w) be a network, that is, a graph G = (V (G), E(G)) together with the weight function w : E(G) → R. The Szeged index Sz(G,w) of the network (G,w) is introduced and proved that Sz(G,w) ≥ W (G,w) holds for any connected network where W (G,w) is the Wiener index of (G,w). Moreover, equality holds if and only if (G,w) is a block network in which w is constant on each of its blocks. Analogous...

We construct several infinite families of trees which have a unique branching vertex of degree 4 and whose Wiener index equals the Wiener index of their quadratic line graph. This solves an open problem of Dobrynin and Mel’nikov.

Let G be a graph. Denote by L(G) its i-iterated line graph and denote by W (G) its Wiener index. Dobrynin, Entringer and Gutman stated the following problem: Does there exist a non-trivial tree T and i ≥ 3 such that W (L(T )) = W (T )? In a series of five papers we solve this problem. In a previous paper we proved that W (L(T )) > W (T ) for every tree T that is not homeomorphic to a path, claw...