Matrix distance 84 -connected

The formerly proposed and the most important of this series of topological indices is the Baiaban distance connectivity index J (also called distance connectivity index or average distance sum connectivity). It is one of the most discriminating - molecular descriptors and its values do not increase substantially with molecule size or number of rings it is defined in terms of sums over each ith row of the - distance matrix D, i.e. the vertex distance degree o [Baiaban, 1982 Baiaban, 1983a]. It is defined as ... 

In order to account for both bond multiplicity and heteroatoms, Balaban modified distance connectivity indices and J were proposed [Balaban, 1986a Balaban et al., 1990a]. These are defined in the same way as the Balaban distance connectivity index but derived from the multigraph distance matrix D instead of the original distance matrix D ... 

Balaban-type index Balaban distance connectivity indices Bartell resonance energy -> resonance indices barycentre centre of mass -> centre of a molecule Barysz index - weighted matrices Barysz distance matrix weighted matrices... 

Other important molecular descriptors obtained from the distance matrix are -> determinant-based descriptors and - Balaban distance connectivity indices. 

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

Balaban distance connectivity index Schultz molecular topological index Kier shape descriptors eigenvalues of the adjacency matrix eigenvalues of the distance matrix Mohar indices Kirchhoff number detour index... 

In analogy with the Kier-Hall - connectivity indices x and the Balaban distance connectivity index J, JJ indices [Randic et al, 1994a] are derived from the Wiener matrix based on the Wiener matrix degrees q, ... 

Calculation of the Balaban distance connectivity index] and Jt index for 2-methylpentane. D is the topological distance matrix distance sums and the vertex degrees. B equals 5 and C is zero. 

Similar to graph potentials, another set of LOVIs was proposed based on the —> geometry matrix G, using as the diagonal terms the —> Balaban distance connectivity index and as the response vector the adjacency matrix A multiplied by the column vector z collecting the atomic numbers of all the non-hydrogen atoms [Beteringhe, Filip et al, 2005] ... 

Topological Distance Measure Coimectivity Indices = Topological Distance Connectivity Indices distance matrix... 

Balaban-like information indices are calculated by replacing vertex distance degrees of the —> Balaban distance connectivity index J with different local invariants that measure the information content of the matrix elements associated with the respective vertex, defined as... 

The variable Balaban index, denoted as J, is calculated by analogy with the Balaban distance connectivity index from the row sums of the variable augmented distance matrix as [Randic and Pompe, 2001b]... 

Based on the cluster-network model, the dependence of current efficiency on the polymer structure was first developed by Gierke [34]. This postulates that clusters with diameters of 4 nm are distributed throughout the matrix and connected to each other by short narrow channels 1 nm in diameter—the cluster separation distance being 5 nm. It should be noted that this structure was developed based on experimental evidence [23]. High caustic current efficiency, according to this model, is a result of the repulsive electrostatic interaction between the OH ions and the fixed ionic charges on the surface... 

An algorithm based on the Laplacian matrix has been proposed for efficacious computing of the resistance-distance matrix for connected graphs (Babid et al., 2002). This computational algorithm consists of the following steps ... 

In 1982 Balaban proposed an index that is analogous to the connectivity index but is based on the graph distance matrix. The connectivity index can be constructed from the adjacency matrix by identifying m and n, the valences of the atoms forming a bond ( n, n), with the row sums for the corresponding atoms. Balaban considered the row sums for the... 

In light of tire tlieory presented above one can understand tliat tire rate of energy delivery to an acceptor site will be modified tlirough tire influence of nuclear motions on tire mutual orientations and distances between donors and acceptors. One aspect is tire fact tliat ultrafast excitation of tire donor pool can lead to collective motion in tire excited donor wavepacket on tire potential surface of tire excited electronic state. Anotlier type of collective nuclear motion, which can also contribute to such observations, relates to tire low-frequency vibrations of tire matrix stmcture in which tire chromophores are embedded, as for example a protein backbone. In tire latter case tire matrix vibration effectively causes a collective motion of tire chromophores togetlier, witliout direct involvement on tire wavepacket motions of individual cliromophores. For all such reasons, nuclear motions cannot in general be neglected. In tliis connection it is notable tliat observations in protein complexes of low-frequency modes in tlie... 

Both the adjacency and distance matrices provide information about the connections in the molceular structure, but no additional information such as atom type or bond order. One type of matrix which includes more information, the Atom Connectivity Matrix (ACM), was introduced by Spialtcr and is discussed in Ref, [38]. This approach was eventually abandoned but is listed here because it was quite a unique approach. 

In a simple (nonweighted) connected graph, the graph distance dy between a pair of vertices V and Vj is equal to the length of the shortest path cormecting the two vertices, i.e. the number of edges on the shortest path. The distance between two adjacent vertices is 1. The distance matrix D(G) of a simple graph G with N vertices is the square NxN symmetric matrix in which [D],j=cl,j [9, 10]. 

Figure 32.8 shows the biplot constructed from the first two columns of the scores matrix S and from the loadings matrix L (Table 32.11). This biplot corresponds with the exponents a = 1 and p = 1 in the definition of scores and loadings (eq. (39.41)). It is meant to reconstruct distances between rows and between columns. The rows and columns are represented by circles and squares respectively. Circles are connected in the order of the consecutive time intervals. The horizontal and vertical axes of this biplot are in the direction of the first and second latent vectors which account respectively for 86 and 13% of the interaction between rows and columns. Only 1% of the interaction is in the direction perpendicular to the plane of the plot. The origin of the frame of coordinates is indicated... 

Balaban-type index from polarizability weighted distance matrix connectivity index chi-0... 

The minimal spanning tree also operates on the distance matrix. Here, near by patterns are connected with lines in such a way that the sum of the connecting lines is minimal and no closed loops are constructed. Here too the information on distances is retained, but the mutual orientation of patterns is omitted. Both methods, hierarchical clustering and minimal spanning tree, aim for making clusters in the multi-dimensional space visible on a plane. 

---