Extended distance matrix

This matrix has been introduced by Tratch, Stankevich, and Zefirov. " In an acyclic graph the elements in this matrix count, for a fixed pair of vertices (/, j), all paths in the graph that contain the path from vertex i to vertex j. The expanded distance matrix has zero diagonal elements, while the element (i, j) is given by the product pij dij, where p-,j is the total number of such paths and dij is the length of the path between i and j. The extended distance matrix for Gi is illustrated in Table 7 where the elements are shown as the products pij dij. 

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. 

EAI index -> eigenvalue-based descriptors (O extended adjacency matrix indices) eccentric distance matrix eccentric connectivity index ( )... 

A group of local vertex invariants and corresponding molecular graph invariants derived from the distance matrix were proposed as quantities related to the molecular complexity [Raychaudhury, Ray et al, 1984, 1992, 1993c] these are vertex complexity, vertex distance complexity, normalized vertex distance complexity, relative vertex distance complexity, mean extended local information on distances, extended local information on distances, Balaban-like information indices, graph vertex complexity, and graph distance complexity. 

Ivanciuc (2000c) extended the concept of reciprocal of the complementary vertex-distance matrix to the vertex- and edge-weighted graphs and used the derived Wiener-like indices in QSPR modeling. 

The usage of matrix representatives of molecules is not restricted to LDMs and can be extended to quantities directly derivable from both theory and experiment such as the matrix of Coulombic nuclear-nuclear repulsion, the distance matrix, or the matrices of bond critical point (BCP) properties such as the electron density-weighted adjacency/connectivity matrix (EDWAM/EDWCM) [22-24, 30]. 

This concept could be extended to any other linear and nonlinear QSAR relationships, by calculating either n x n distance matrices D (especially suited for nonlinear relationships) or n X n covariance matrices C as similarity measures. For this purpose, all or only several relevant properties of the compounds are used to calculate the corresponding similarity matrices. No superposition of the molecules is necessary. If a distance matrix D is calculated from the X matrix of explanatory physicochemical properties n rows, m columns), then all Xij values must be normalized before, i.e., mean-value-centered and standardized, column by column. The great advantage of distance similarity index matrices is that no special models need to be defined in the case of nonlinear relationships on the other hand, problems may arise from significant intercorrelations between the different columns of the similarity matrices. 

The distance between object points is considered as an inverse similarity of the objects. This similarity depends on the variables used and on the distance measure applied. The distances between the objects can be collected in a distance matrk. Most used is the euclidean distance, which is the commonly used distance, extended to more than two or three dimensions. Other distance measures (city block distance, correlation coefficient) can be applied of special importance is the mahalanobis distance which considers the spatial distribution of the object points (the correlation between the variables). Based on the Mahalanobis distance, multivariate outliers can be identified. The Mahalanobis distance is based on the covariance matrix of X this matrix plays a central role in multivariate data analysis and should be estimated by appropriate methods—mostly robust methods are adequate. 

In most cases, however, polymers crystallize neither completely nor perfectly. Instead, they give semicrystalline materials, containing crystalline regions separated by adjacent amorphous phases. Moreover, the ordered crystalline regions may be disturbed to some extent by lattice defects. The crystalline regions thus embedded in an amorphous matrix typically extend over average distances of 10-40 nm. The fraction of crystalline material is termed the degree of crystallinity. This is an important parameter of semicrystalline materials. 

For example, consider the crack tip as it intersects a fiber (Fig. 16). The local stresses at the tip can cause fiber-matrix debonding. The crack tip continues to open causing the interfacial debonded region to extend. The fiber continues to interact with the matrix through a frictional sliding force even after the initial bond fails. The distance over which the force acts is the debonded length times the difference in strain between the fiber and the matrix. 

So far we have taken the tunneling matrix element A0 to be independent of vibrational coordinates. In terms of our original model with extended tunneling coordinate Q, this assumption means that the vibrations asymmetrize the instantaneous potential V(Q, < , ) but do not modulate its height or width. This model does not describe the effect of vibration on tunneling (fluctuational barrier preparation) dealt with in Section 2.5. For example, consider the OH O fragment shown in Figure 1.2. The relative 0-0 distance is clearly the same in the initial and final states, and hence the 0-0 vibration cannot be considered linearly coupled to the reaction coordinate. Such a mode (call it qx) is not... 

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