The Multiplicative Version of the Edge Wiener Index (original) (raw)
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Molecular graphs and the inverse Wiener index problem
Discrete Applied Mathematics, 2009
In the drug design process, one wants to construct chemical compounds with certain properties. In order to establish the mathematical basis for the connections between molecular structures and physicochemical properties of chemical compounds, some so-called structure-descriptors or "topological indices" have been put forward. Among them, the Wiener index is one of the most important. A long standing conjecture on the Wiener index ([6], [9]) states that for any positive integer n (except numbers from a given 49 element set), one can find a tree with Wiener index n. We proved this conjecture in [13] and [14]. However, more realistic molecular graphs are trees with degree ≤ 3 and the so-called hexagon type graphs. In this paper, we prove that every sufficiently large integer n is the Wiener index of some caterpillar tree with degree ≤ 3, and every sufficiently large even integer is the Wiener index of some hexagon type graph.
The Multiplicative Version of the Wiener Index
The classical Wiener index, W(G), is equal to the sum of the distances between all pairs of vertexes of a (molecular) graph, G. We now consider a related topological index, π(G), equal to the product of distances between all pairs of vertexes of G. The basic properties of the π index are established and its possible physicochemical applications examined. In the case of alkanes, π and W are highly correlated; a slightly curvilinear correlation exists between ln π and W.
2017
The cut method demonstrates its usefulness especially for the topological indices that are based on the distances in the molecular graphs without actually calculating the distances between pairs of vertices. The Wiener index is equal to the sum of distances between all pairs of vertices of the connected graph G, whereas the Edge-Wiener index is the sum of distances between all pairs of edges of the connected graph G. In this paper we calculate the Edge-Wiener indices of Circum-polyacenes, Circum-pyrenes and Circum-trizenes. MSC: 05C12, 05C90, 05C76.
Wiener Index of Degree Splitting Graph of some Hydrocarbons
International Journal of Computer Applications, 2014
In quantum chemistry, the physico-chemical properties of chemical compounds are often modeled by means of molecular-graph-based structure-descriptors, which are also referred to as topological indices. One of the most widely known topological descriptors is Wiener index. It is named after chemist Harold Wiener who introduced in the year 1947. It is defined by the sum of the distances between all (ordered) pairs of vertices of G. In this paper, we find the Wiener index of degree splitting of some aliphatic and aromatic hydrocarbons and classify its characterization using MATLAB.
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.
On the Wiener Index of FH Sums of Graphs
2021
Wiener index is the first among the long list of topological indices which was used to correlate structural and chemical properties of molecular graphs. In [5] M. Eliasi, B. Taeri defined four new sums of graphs based on the subdivision of edges with regard to the cartesian product and computed their Wiener index. In this paper, we define a new class of sums called FH sums and compute the Wiener index of the resulting graph in terms of the Wiener indices of the component graphs so that the results in [5] becomes a particular case of the Wiener index of FH sums for H = K1, the complete graph on a single vertex.
Wiener index of Eulerian graphs
Discrete Applied Mathematics, 2014
The forgotten topological index of a molecular graph is defined as F(G) = ∑ () ∈ () , where () denotes the degree of vertex in. The first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number = 1, 2, the first through the fourth for = 3, and the first and the second for = 4, 5 are determined. These results are compared with those obtained for the first Zagreb index.
On the terminal wiener index of thorn graphs
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 distancebased molecular structure descriptor was put forward quite recently [I. Gutman, B. Furtula, M. Petrovic, J. ...