Wiener Polarity Index Research Articles

The number of independent sets in a connected graph and its complement

For a connected graph G, the total number of independent vertex sets (including the empty vertex set) is denoted by i(G). In this paper, we consider Nordhaus-Gaddum-type inequalities for the number of independent sets of a connected graph with connected complement. First we define a transformation on a graph that increases i(G) and i(G). Next, we obtain the minimum and maximum value of i(G) + i(G), where graph G is a tree T with connected complement and a unicyclic graph G with connected complement, respectively. In each case, we characterize the extremal graphs. Finally, we establish an upper bound on the i(G) in terms of the Wiener polarity index.

A note on chemical trees with minimum Wiener polarity index

The Wiener polarity index (usually denoted by Wp) of a graph G is defined as the number of unordered pairs of the vertices of G which are at distance 3. Denote by CTn the family of all n-vertex chemical trees. In a recent paper, Ashrafi and Ghalavand [1] determined the first three minimum Wp values of n-vertex chemical trees for n ≥ 7 and characterized the chemical trees attaining the first two minimum Wp values among all the members of CTn for n ≥ 4. In this note, the chemical trees with the third minimum Wp value are characterized from the graph family CTn for n ≥ 7, and the chemical trees from the family CTn,n ≥ 4, with the first two minimum Wp values are also obtained in an alternative but shorter way.

The inverse Wiener polarity index problem for chemical trees.

The Wiener polarity number (which, nowadays, known as the Wiener polarity index and usually denoted by Wp) was devised by the chemist Harold Wiener, for predicting the boiling points of alkanes. The index Wp of chemical trees (chemical graphs representing alkanes) is defined as the number of unordered pairs of vertices (carbon atoms) at distance 3. The inverse problems based on some well-known topological indices have already been addressed in the literature. The solution of such inverse problems may be helpful in speeding up the discovery of lead compounds having the desired properties. This paper is devoted to solving a stronger version of the inverse problem based on Wiener polarity index for chemical trees. More precisely, it is proved that for every integer t ∈ {n − 3, n − 2,…,3n − 16, 3n − 15}, n ≥ 6, there exists an n-vertex chemical tree T such that Wp(T) = t.

Wiener polarity index of quasi-tree molecular structures

Wiener polarity index of quasi-tree molecular structures

On the generalized Wiener polarity index of trees with a given diameter

The generalized Wiener polarity index of a graph G, denoted by Wk(G), is the number of unordered pairs of vertices that are at distance k in G. In this paper, we characterize the extremal trees with respect to the index among all trees of order n and diameter d, which partially answers a question of Bollobás and Tyomkyn (2012) and also generalizes some results of Deng et al. (2010).

Analyzing lattice networks through substructures

Analyzing the topology of network structures is an important topic studied from many different aspects of science and mathematics. The Wiener polarity index (number of unordered pairs of vertices at distance 3 from each other) is one of the representative descriptors of graph structures. It was computed for several lattice networks by Chen et al. [11] in an effort to understand the properties of these networks. The Wiener polarity index is a variation of the classic distance-based graph invariant, the Wiener index (sum of distances between all pairs of vertices), which is known to be closely related to the number of substructures. In this paper we examine the numbers of various subgraphs of order 4 for these lattice networks. In addition to confirming their symmetric nature, comparing the numbers of various substructures leads to insights on other less trivial characteristics of these network structures of common interest.

Relationships between some distance-based topological indices

The Harary index (HI), the average distance (AD), the Wiener polarity index (WPI) and the connective eccentricity index (CEI) are distance-based graph invariants, some of which found applications in chemistry. We investigate the relationship between HI, AD, and CEI, and between WPI, AD, and CEI. First, we prove that HI > AD for any connected graph and that HI > CEI for trees, with only three exceptions. We compare WPI with CEI for trees, and give a classification of trees for which CEI ? WPI or CEI < WPI. Furthermore, we prove that for trees, WPI > AD, with only three exceptions.

The Wiener Polarity Index of Benzenoid Systems and Nanotubes

In this paper, we consider a molecular descriptor called the Wiener polarity index, which is defined as the number of unordered pairs of vertices at distance three in a graph. Molecular descriptors play a fundamental role in chemistry, materials engineering, and in drug design since they can be correlated with a large number of physico-chemical properties of molecules. As the main result, we develop a method for computing the Wiener polarity index for two basic and most commonly studied families of molecular graphs, benzenoid systems and carbon nanotubes. The obtained method is then used to find a closed formula for the Wiener polarity index of any benzenoid system. Moreover, we also compute this index for zig-zag and armchair nanotubes.

Determination of wiener polarity index from reduced distance matrix and molecular structure of drugs-A theoretical approach

Numerous studies indicate that drug properties like boiling point, melting point, pKa value etc., indicate that there is a strong relationship between chemical characteristics of chemical compounds and drugs and their molecular structures. In the fields of chemical graph theory, molecular topology and mathematical chemistry, a topological index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological descriptors are derived from hydrogen-suppressed molecular graphs, in which the atoms are represented by vertices and the bonds by edges. The connections between the atoms can be described by various types of topological matrices like a distance or adjacency matrix, which can be mathematically manipulated so as to derive a single number, usually known as graph invariant, graph-theoretical index or topological index. Smaller the size of the matrix, the easier is the computation and hence better the runtime of the algorithms developed. This presentation aims to introduce a technique to determine the Wiener polarity index of a tree from its subtree, which aids in reducing the size of the distance matrix used for calculating the Wiener polarity index and hence reduces the computational complexity.

The Wiener index of Sierpiński-like graphs

Sierpinski-like graphs constitute an extensively studied family of graphs of fractal nature applicable in topology, mathematics of the Tower of Hanoi, computer science, and elsewhere. In this paper, we focus on the Wiener polarity index, Wiener index and Harary index of Sierpinski-like graphs. By Sierpinski-like graphs’ special structure and correlation, their Wiener polarity index and some Sierpinski-like graph’s Wiener index and Harary index are obtained.

Wiener polarity index of dendrimers

Network structures have many applications in biological, information theory, physical and social sciences. Research on network measurements is more and more important recently. A number of topological indices have been studied as descriptors of complex networks. The Wiener polarity index has been introduced for a long time and known to be related to the cluster coefficient of networks. In this paper, we study the Wiener polarity index of dendrimers, which has many applications in other fields.

Ordering chemical trees by Wiener polarity index

For a molecular graph G with vertex set V(G), the Wiener polarity index Wp(G) is the number of unordered pairs of vertices {u, v} such that dG(u,v)=3. In this paper, an ordering of chemical trees of order n with respect to Wiener polarity index is given.

On the Wiener Polarity Index of Lattice Networks.

Network structures are everywhere, including but not limited to applications in biological, physical and social sciences, information technology, and optimization. Network robustness is of crucial importance in all such applications. Research on this topic relies on finding a suitable measure and use this measure to quantify network robustness. A number of distance-based graph invariants, also known as topological indices, have recently been incorporated as descriptors of complex networks. Among them the Wiener type indices are the most well known and commonly used such descriptors. As one of the fundamental variants of the original Wiener index, the Wiener polarity index has been introduced for a long time and known to be related to the cluster coefficient of networks. In this paper, we consider the value of the Wiener polarity index of lattice networks, a common network structure known for its simplicity and symmetric structure. We first present a simple general formula for computing the Wiener polarity index of any graph. Using this formula, together with the symmetric and recursive topology of lattice networks, we provide explicit formulas of the Wiener polarity index of the square lattices, the hexagonal lattices, the triangular lattices, and the 33 ⋅ 42 lattices. We also comment on potential future research topics.

On the Wiener polarity index of graphs

The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u, v} in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied the Wiener polarity index in [Y. Zhang, Y. Hu, 2016] [38]. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus–Gaddum-type results for Wp(G). Our lower bound on Wp(G)+Wp(G¯) is always better than the previous lower bound given by Zhang and Hu.

On Wiener polarity index of bicyclic networks

Complex networks are ubiquitous in biological, physical and social sciences. Network robustness research aims at finding a measure to quantify network robustness. A number of Wiener type indices have recently been incorporated as distance-based descriptors of complex networks. Wiener type indices are known to depend both on the network’s number of nodes and topology. The Wiener polarity index is also related to the cluster coefficient of networks. In this paper, based on some graph transformations, we determine the sharp upper bound of the Wiener polarity index among all bicyclic networks. These bounds help to understand the underlying quantitative graph measures in depth.

On the Extremal Wiener Polarity Index of Hückel Graphs.

Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. The Wiener polarity index W p(G) of a graph G is the number of unordered pairs of vertices u, v of G such that the distance between u and v is equal to 3. The trees and unicyclic graphs with perfect matching, of which all vertices have degrees not greater than three, are referred to as the Hückel trees and unicyclic Hückel graphs, respectively. In this paper, we first consider the smallest and the largest Wiener polarity index among all Hückel trees on 2n vertices and characterize the corresponding extremal graphs. Then we obtain an upper and lower bound for the Wiener polarity index of unicyclic Hückel graphs on 2n vertices.

The Nordhaus–Gaddum-type inequality for the Wiener polarity index

The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u, v} of G such that the distance between u and v is 3. In this paper, we study the Nordhaus–Gaddum-type inequality for the Wiener polarity index of a graph G of order n. Due to concerns that both Wp(G) and Wp(G¯) are nontrivial only when diam(G)=3 and diam(G¯)=3, we firstly consider the crucial case and get that 2≤Wp(G)+Wp(G¯)≤⌈n2⌉⌊n2⌋−n+2. Moreover, the bounds are best possible, and the corresponding extremal graphs are also presented. Then we generalize the results to all connected simple graphs. Furthermore, we discuss the Nordhaus–Gaddum-type inequality for trees and unicyclic graphs, respectively.

Highly Correlated Wiener polarity Index: A Model to Predict Log p

Wiener polarity index of a more generalized graph of chemical graphs of polyacenes and phenylenes is computed and the results are used to design a model for predicting log p values of polyacenes. The multiple regression with two descriptors gives improved models with correlation coefficient 0.999999980642502.

Wiener Polarity Index of Cycle-Block Graphs

The Wiener polarity index WP of a graph G is the number of unordered pairs of vertices u,v of G such that the distance dG(u, v) between u and v is 3. Cycle-block graph is a connected graph in which every block is a cycle. In this paper, we determine the maximum and minimum Wiener polarity index of cycle-block graphs and describe their extremal graphs; the extremal graphs of 4-uniform cactus with respect to Wiener polarity index are also discussed.

On Minimum Wiener Polarity Index of Unicyclic Graphs with Prescribed Maximum Degree

The Wiener polarity index of a connected graphGis defined as the number of its pairs of vertices that are at distance three. By introducing some graph transformations, in different way with that of Huang et al., 2013, we determine the minimum Wiener polarity index of unicyclic graphs with any given maximum degree and girth, and characterize extremal graphs. These observations lead to the determination of the minimum Wiener polarity index of unicyclic graphs and the characterization of the extremal graphs.