On the Wiener Index of Some Total Graphs

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.

2021

Topological indices are graph invariants computed usually by means of the distances or degrees of vertices of a graph. In chemical graph theory, a molecule can be modeled by a graph by replacing atoms by the vertices and bonds by the edges of this graph. Topological graph indices have been successfully used in determining the structural properties and in predicting certain physicochemical properties of chemical compounds. Wiener index is the oldest topological index which can be used for analyzing intrinsic properties of a molecular structure in chemistry. The Wiener index of a graph $G$ is equal to the sum of distances between all pairs of vertices of $G$. Recently, the entire versions of several indices have been introduced and studied due to their applications. Here we introduce the entire Wiener index of a graph. Exact values of this index for trees and some graph families are obtained, some properties and bounds for the entire Wiener index are established. Exact values of this ...

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.

In this paper, we define new graph operations F-composition F(G)[H], where F(G) be one of the symbols S(G), M(G), Q(G), T (G), Λ(G), Λ[G], D 2 (G), D 2 [G]. Further, we give some results for the Wiener indices of the these graph operations.

2010

The Wiener index of a graph G, denoted by W (G), is the sum of distances between all pairs of vertices in G. In this paper, we consider the relation between the Wiener index of a graph, G, and its line graph, L(G). We show that if G is of minimum degree at least two, then W (G) ≤ W (L(G)). We prove that for every non-negative integer g 0 , there exists g > g 0 , such that there are infinitely many graphs G of girth g, satisfying W (G) = W (L(G)). This partially answers a question raised by Dobrynin and Mel'nikov [8] and encourages us to conjecture that the answer to a stronger form of their question is affirmative.

If G is a simple graph, then con(G), the common neighborhood graph of G, has the same vertex set as G, and two vertices of con(G) are adjacent if they have a common neighbor in G. We show that for any bipartite graph G the Wiener index (i.e., sum of distances between all pairs of vertices) of con(G) is always smaller than the Wiener index of G. For general graphs, however, the Wiener index of common neighbor graphs can be greater. This fact is surprising, since we also show that the diameter of con(G) is at most the diameter of G. We present constructions of two infinite classes of graphs, chemical and unicyclic graphs, which have this property.

Applied Mathematics Letters, 2007

The sum of distances between all vertex pairs in a connected graph is known as the Wiener index. It is an early index which correlates well with many physico-chemical properties of organic compounds and as such has been well studied over the last quarter of a century. A q-analogue of this index, termed the Wiener polynomial by Hosoya but also known today as the Hosoya polynomial, extends this concept by trying to capture the complete distribution of distances in the graph.

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.

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 bipartite graph is called a chain graph if the neighborhoods of the vertices in each partite set form a chain with respect to set inclusion. Chain graphs are discovered and rediscovered by various researchers in various contexts. Also, they were named differently according to the applications in which they arise. In the field of Spectral Graph Theory, chain graphs play a remarkable role. They are characterized as graphs with the largest spectral radius among all the connected bipartite graphs with prescribed number of edges and vertices. Even though chain graphs are significant in the field of Spectral Graph Theory, the area of graph parameters remains untouched. Wiener index is one of the oldest and most studied topological indices, both from theoretical point of view and applications. The Wiener index of a graph is defined as the sum of distances between every pair of vertices in it. In this article, we give bounds for Wiener index and hence for average distance of chain graphs. We also present a quadratic time algorithm that returns a realising chain graph G (if exists) whose Wiener index is the given integer w. The algorithm presented in this article contributes to the existing knowledge in the theory of inverse Wiener index problem. We conclude this article by exploring some more graph parameters called edge Wiener index, hyper Wiener index and Zagreb indices of chain graphs.