Computing Wiener Index of Fibonacci Weighted Trees (original) (raw)

Wiener index of trees: theory and applications

Acta Applicandae Mathematicae, 2001

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism-discriminating power of W , connections between W and the center and centroid of a tree, as well as between W and the Laplacian eigenvalues, results on the Wiener indices of the line graphs of trees, on trees extremal w.r.t. W , and on integers which cannot be Wiener indices of trees. A few conjectures and open problems are mentioned, as well as the applications of W in chemistry, communication theory and elsewhere. : Primary 05C12, 05C05; Secondary 05C90.

A New Approach to Compute Wiener Index

Journal of Computational and Theoretical Nanoscience, 2013

Distance properties of molecular graphs form an important topic in chemical graph theory. The Wiener index of a graph G is defined as the sum of all distances between distinct vertices of G. A lot of research has been devoted to finding Wiener index by brute force method. In this paper we develop a method to compute the Wiener index of certain chemical graphs without using distance matrix.

Wiener index versus maximum degree in trees

Discrete Applied Mathematics, 2002

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. ?

Wiener index, number of subtrees, and tree eccentric sequence

arXiv (Cornell University), 2020

The eccentricity of a vertex u in a connected graph G is the distance between u and a vertex farthest from it; the eccentric sequence of G is the nondecreasing sequence of the eccentricities of G. In this paper, we determine the unique tree that minimises the Wiener index, i.e. the sum of distances between all unordered vertex pairs, among all trees with a given eccentric sequence. We show that the same tree maximises the number of subtrees among all trees with a given eccentric sequence, thus providing another example of negative correlation between the number of subtrees and the Wiener index of trees. Furthermore, we provide formulas for the corresponding extreme values of these two invariants in terms of the eccentric sequence. As a corollary to our results, we determine the unique tree that minimises the edge Wiener index, the vertex-edge Wiener index, the Schulz index (or degree distance), and the Gutman index among all trees with a given eccentric sequence.

How to compute the Wiener index of a graph

Journal of Mathematical Chemistry, 1988

The Wiener index of a graph G is equal to the sum of distances between all pairs of vertices of G, It is known that the Wiener index of a molecular graph correlates with certain physical and chemical properties of a molecule. In the mathematical literature, many good algorithms can be found to eompute the distances m a graph, and these can easily be adapted for the caleulation of the Wiener indcx. An algorithm that calculates the Wiener index of a tree in linear timc is givcn. It improves an algorithm of Canfield, Robinson and Rouvray. The question rcmains: is there an algorithm for general graphs that would eMculate the Wiener index without calculating the distance matrix? Another algorithm that calculates this index for an arbitrary graph is given.

Wiener Index of Graphs using Degree Sequence

2012

The Wiener index of a graph is defined as the sum of distances between all pairs of vertices in a connected graph. Wiener index correlates well with many physio chemical properties of organic compounds and as such has been well studied over the last quarter of a century. In this paper we prove some general results on Wiener Index for graphs using degree sequence.

Variation of the Wiener index under tree transformations

Discrete Applied Mathematics, 2005

The Wiener index W (T) is defined as the sum of distances between all pairs of vertices of the tree T. In this paper we find the variation of the Wiener index under certain tree transformations, which can be described in terms of coalescence of trees. As a consequence, conditions for nonisomorphic trees having equal Wiener index are presented. Also, a partial order on the collection of trees (with a fixed number of vertices) is introduced, providing structural information about the behavior of W.

Bounding the kkk-Steiner Wiener and Wiener-Type Indices of Trees in Terms of Eccentric Sequence

Acta Applicandae Mathematicae, 2021

The eccentric sequence of a connected graph G is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of G is the sum of the distances between all unordered pairs of vertices of G. The unique trees that minimise the Wiener index among all trees with a given eccentric sequence were recently determined by the present authors. In this paper we show that these results hold not only for the Wiener index, but for a large class of distance-based topological indices which we term Wiener-type indices. Particular cases of this class include the hyper-Wiener index, the Harary index, the generalised Wiener index W λ for λ > 0 and λ < 0, and the reciprocal complementary Wiener index. Our results imply and unify known bounds on these Wiener-type indices for trees of given order and diameter. We also present similar results for the k-Steiner Wiener index of trees with a given eccentric sequence. The Steiner distance of a set A ⊆ V (G) is the minimum number of edges in a subtree of G whose vertex set contains A, and the k-Steiner Wiener index is the sum of distances of all k-element subsets of V (G). As a corollary, we obtain a sharp lower bound on the k-Steiner Wiener index of trees with given order and diameter, and determine in which cases the extremal tree is unique, thereby correcting an error in the literature.

Wiener Index of Directed and Weighted Graphs by MATLAB Program

Abstract: The Wiener index is the one of the oldest and most commonlyused topological indices in the quantitative structure-property relationships. It is defined by the sum of the distances between all (ordered) pairs of vertices of G. In this paper, we use MATLAB program for finding the Wiener index of the vertex weighted, edge weighted directed and undirected graphs Keywords: Distance Sum, MATLAB, Sparse Matrix, Wiener Index