Index Wiener

Wiener Index The Wiener index, W, is defined as the sum of topological distances in G [15], considering either the upper right or lower left triangle of D  

The different structural parameters such as degree of polymerization, degree of branching and Wiener index can be used to characterize the topologies of hypeibranched polymers. The degree of branching (DB) is defined as follows  

Where D is the number of dendritic units and L is the number of linear units. This value varies from 0 for linear polymers to 1 for dendrimers or fully branched hyper-branched polymers [26-28]. 

In addition to the degree of branching, the Wiener index is also used to distinguish polymers of different topologies and defined as  

Where N is the number of beads per molecule and d. is the number of bonds separating site / and j of the molecule. This parameter only describes the connectivity and is not a direct measure of the size of the molecules. Larger Wiener index numbers indicate higher numbers of bonds separating beads in molecules and hence more open structures of polymer molecules [28]. Table 2 shows the DB of polymers with different architectnre and the same degree of polymerization. 

TABLE 2 Degree of branching for different polymer architectures of the same molecular weight (white beads representing linear units and gray beads representing branching units). 

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Hence, they take account of only the structure constitution (topology). One of the first and most frequently used topological indices is the Wiener index. It has the form of Eq. (3), where D,y are all the routes from atom i to atom j. 

Z eb index, Wiener index. Balaban J index, connectivity indices chi (x), kappa (k) shape indices, molecular walk counts, BCUT descriptors, 2D autocorrelation vector... 

For historical reasons the Wiener index, W, is introduced in this section. It was defined in 1947 and is still a starting point for the invention of new topological indices. 

The Wiener index was originally defined only for acyclic graphs and was initially called the path number [6]. "The path number, W, is defined as the sum of the distances between any two carbon atoms in the molecule in terms of carbon-carbon bonds". Hosoya extended the Wiener index and defined it as the half-sum of the off diagonal elements of a distance matrix D in the hydrogen-depleted molecular graph of Eq, (15), where dij is an element of the distance matrix D and gives the shortest path between atoms i and j. 

Because of the symmetry of the distance matrix, the Wiener index can be expressed as Eq. (16). 

With Eq. (16) the Wiener index of compound 2 can be calculated from the distance matrix as shown in Eq, (17)... 

W Wiener index — half-sum of the off-diagonal elements of the distance matrix of a graph... 

Obviously, R = 2w, similarly to A = 2A for the total adjacency. Proceeding from this analogy one can call either w or R the total distance of the graph, giving a preference to the Wiener index since, due to the symmetry relative to the main matrix diagonal, the Rouvray index is redundant. Applications of w = R/2 are discussed in the following section. 

This is another index, which was independently developed by Polansky 46) for expressing some graph properties, including the relation with the Wiener index for important classes of molecular graphs, and by Bonchev, Balaban and Mekenyan 47)... 

Wiener-Index-AHvb Relationship Bonchev et al. [13] have reported the following relationship for alkanes (C2-C10) ... 

Fig. 12.3 Minimum Wiener-index principle Wiener Index is able to select compact isomers, chemically stable...

I. Gutman et al., The graph of atomic orbitals and its basic properties. 1. Wiener index. MATCH Commun. Math. Comput. Chem. 53, 215-224 (2005)... 

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