Wiener Index of Directed and Weighted Graphs by MATLAB Program

2017

A Hosoya polynomial is a polynomial connected to a molecular graph, which is a graph representation of a chemical compound with atoms as vertices and chemical bonds as edges. A graph invariant is the Hosoya polynomial; it is a graph attribute that does not change under graph isomorphism. It provides information about the number of unique nonempty subgraphs in a given graph. A molecular graph's size and branching complexity are determined by a topological metric known as the Wiener index. The Wiener index of each pair of vertices in a molecular network is the sum of those distances. The topological index, one of the various classes of graph invariants, is a real number related to a connected graph's structure .The goal of this article is to compute the Hosoya polynomial of some class of Abid-Waheed graph. Further, this research focused on a C++ algorithm to calculate the wiener index of AW 9 m and AW 11 m . The Wiener index (W * I) and Hyper-Wiener index (H * W * I) are calculated using Hosoya polynomial (H * -polynomial) of some family of Abid-Waheed graphs AW 9 m and AW 11 m . Illustrations and applications are given to enhance the research work.

2011

a b s t r a c t Eliasi and Taeri [Extension of the Wiener index and Wiener polynomial, Appl. Math. Lett. 21 (2008) 916-921] introduced the notion of y-Wiener index of graphs as a generalization of the classical Wiener index and hyper Wiener index of graphs. They obtained some mathematical properties of this new defined topological index. In this paper, the join, Cartesian product, composition, disjunction and symmetric difference of graphs under y-Wiener index are computed. By these results most parts of a paper by Sagan et al. [The Wiener polynomial of a graph, Int. J. Quant. Chem. 60 (1996) 959-969] and another paper by Khalifeh et al. [The hyper-Wiener index of graph operations, Comput. Math. Appl. 56 (2008) 1402-1407] are generalized.

Kragujevac Journal of Science, 2010

ABSTRACT. The terminal Wiener index TW= TW (G) of a graph G is equal to the sum of distances between all pairs of pendent vertices of G. This distance–based molecular structure descriptor was put forward quite recently [I. Gutman, B. Furtula, M. Petrovic, J. ...

Proceedings of the 2nd Croatian Combinatorial Days, 2019

The Wiener index W (G) of a connected graph G is defined as the sum of distances between all pairs of vertices in G. In 1991,Šoltés [9] posed the problem of finding all graphs G such that equality W (G) = W (G − v) holds for all vertices v in G. The only known graph with this property is the cycle C 11. Our main object of study is the relaxed version of this problem: find graphs for which Wiener index does not change when a particular vertex v is removed. This overview contains results which were obtained and published during the past two years concerning relaxedŠoltés's problem.

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.

PLoS ONE, 2014

In this paper, we derive interrelations of graph distance measures by means of inequalities. For this investigation we are using graph distance measures based on topological indices that have not been studied in this context. Specifically, we are using the well-known Wiener index, Randić index, eigenvalue-based quantities and graph entropies. In addition to this analysis, we present results from numerical studies exploring various properties of the measures and aspects of their quality. Our results could find application in chemoinformatics and computational biology where the structural investigation of chemical components and gene networks is currently of great interest.

Ars Mathematica Contemporanea, 2015

Graovac and Pisanski [On the Wiener index of a graph, J. Math. Chem. 8 (1991) 53 -62] applied an algebraic approach to generalize the Wiener index by symmetry group of the molecular graph under consideration. In this paper, exact formulas for this graph invariant under some graph operations are presented.

Abstract: One of the most widely known topological descriptors is the Wiener index or (Wiener number) named after American chemist Harold Wiener in 1947. Wiener number of a connected graph G is defined as the sum of the distances between distinct pairs of vertices of G..It correlates between physico- chemical and structural properties.The hyper wiener index denoted by WW of a graph G was introduced by Randic and his definition is applicable to trees only. Klein, Lukovits and Gutman introduced the formula for both trees and cycle containing structures. The Hyper Wiener index is defined as WW (G) = (Σd2(u ,v)+Σd(u,v))/2, where d(u,v) denotes the distance between the vertices u and v in the graph G and the summations run over all distinct pairs of vertices of G. Recently an edge version of Hyper Wiener Index was introduced by Ali Iranmanesh.In this paper, we have determined Hyper Wiener numbers of some Cluster graphs and also for some bipartite cluster graphs Keywords: distance sum ,nanostar, Vertex and edge Wiener index, Vertex and edge Hyper Wiener index