Peripheral Wiener Index of Graph Operations

2009

Let d (G, k) be the number of pairs of vertices of a graph G that are at distance k, λ a real number, and Wλ (G)= Σk≥ 1 d (G, k) kλ. Wλ (G) is called the Wiener-type invariant of G associated to real number λ. In this paper, the Wiener-type invariants of some graph operations are computed. As immediate consequences, the formulae for reciprocal Wiener index, Harary index, hyper-Wiener index and Tratch-Stankevich-Zefirov index are calculated. Some upper and lower bounds are also presented.

2018

The peripheral Wiener index, PW (G) is the sum of the distances of all pairs of vertices in the periphery of a graph G. In this paper it is shown that, an arbitrary graph G is an induced subgraph of a graph H for which PW (H) = PW (L(H)) holds, where L(K) is the line graph of K. Using Pell-like equations, infinite families of graphs G are constructed for which PW (G) = PW (L(G)) holds. A connection between the peripheral Wiener index and the Zagreb index for graphs of small diameter is presented. It is also demonstrated that the partition distance approach applicable to the peripheral Wiener index, making some earlier results as very special cases of this approach.

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.

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.

2017

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.

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.

Creative Mathematics and Informatics, 2017

Given a simple and connected graph G with vertex set V , denoting by d G (u) the degree of a vertex u and d G (u, v) the distance of two vertices, the modified Schultz index of G is given by S * (G) = {u,v}⊆V d G (u) d G (v) d G (u, v), where the summation goes over all non ordered pairs of vertices of G. In this paper we consider some graph operations, namely cartesian product, complete product, composition and subdivision, and we obtain explicit formulae for the modified Schultz index of a graph in terms of the number of vertices and edges as well as some other topological invariants such as the Wiener index, the Schultz index and the first and second Zagreb indices.

2014

Abstract. The Wiener index is one of the oldest molecular-graph-based structure-descriptors. It was first proposed by American Chemist Harold Wiener in 1947 as an aid to determining the boiling point of paraffin. The study of Wiener index is one of the current areas of research in mathematical chemistry. It also gives good correlations between Wiener index (of molecular graphs) and the physico-chemical properties of the underlying organic compounds. That is, the Wiener index of a molecular graph provides a rough measure of the compactness of the underlying molecule. The Wiener index W(G) of a connected graph G is the sum of the distances between all pairs (ordered) of vertices of G. ∑ , , . In this paper, we give theoretical results for calculating the Wiener index of a cycle in the context of some graph operations. These formulas will pave the way to demonstrate the Wiener index of molecular structures.

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