The Wiener index of a graph G is defined as the sum of all distances between distinct vertices of G. In this paper an algorithm for constructing distance matrix of a zig-zag polyhex nanotube is introduced. As a consequence, the Wiener index of this nanotube is computed.

In this paper, we investigate how the Wiener index of unicyclic graphs varies with graph operations. These results are used to present a sharp lower bound for the Wiener index of unicyclic graphs of order n with girth and the matching number β ≥ 3g 2 . Moreover, we characterize all extremal graphs which attain the lower bound.

The Wiener index of a graph, which is the sum of the distances between all pairs of vertices, has been well studied. Recently, Sills and Wang in 2012 proposed two conjectures on the maximal Wiener index of trees with a given degree sequence. This note proves one of the two conjectures and disproves the other.

If G is a connected graph with vertex set V , then the eccentric connectivity index of G, ξ(G) is defined as ∑ deg(v).ec(v) where deg(v) is the degree of a vertex v and ec(v) is its eccentricity. The Wiener index W (G) = 1 2 [ ∑ d(u, v)], the hyper-Wiener index WW (G) = 1 2 [ ∑ d(u, v) + ∑ d(u, v)] and the reverseWiener index ∧(G) = n(n−1)D 2 −W (G), where d(u, v) is the distance of two vertice...

We show that graph equation W (L(T )) = W (T ), where T is a tree, W (T ) its Wiener index and L(T ) its line graph, has infinitely many nonhomeomorphic solutions among open quipus. This gives a positive answer to the 2004 problem of Dobrynin and Mel’nikov on the existence of solutions with arbitrarily large number of arbitrarily long pendant paths, and disproves the 2014 conjecture of Knor and...

It is a known fact that the Wiener index (i.e. the sum of all distances between pairs of vertices in a graph) of a tree with an odd number of vertices is always even. In this paper, we consider the distribution of the Wiener index and the related tree parameter “internal path length” modulo 2 by means of a generating functions approach as well as by constructing bijections for plane trees.

The Wiener index W (G) of a connected graph G is defined to be the sum

The Wiener index of a connected graph is the sum of all pairwise distances of vertices of the graph. In this paper, we consider the Wiener indices of trees with perfect matchings, characterizing the eight trees with smallest Wiener indices among all trees of order 2 ( 11) m m with perfect matchings.

The Wiener matrix and the hyper-Wiener number of a tree (acyclic structure), higher Wiener numbers of a tree that can be represented by a Wiener number sequence W, W,W.... whereW = W is the Wiener index, and R W k K    ,.... 2 , 1 is the hyper-Wiener number. The concepts of the Wiener vector and hyper-Wiener vector of a graph are introduced for the molecular graph of bi-phenylene. Moreover, ...