Generalized Distance Matrices

In Table 8.1, we have listed as novel three distance-based matrices the square root distance matrix, the -root distance matrix, and the distance squared matrix, but not because they are new in content and not resembling something already known or that they are original in conception. They are certainly not new because they have been already used in physics and mathematics. However, because they appear to be of potential interest for chemistry and are unknown in chemical literature, one may consider them as a novelty for chemistry. In Table 8.3, we have illustrated the square root distance matrix and the n-root distance matrix, again for the case of 3-methylhexane and norbomane, a bicyclic also seven carbon atom structure. We start with the square root and n-root distance matrix and will discuss the distance squared matrix later. 

The Harary index received attention in both ch ical and mathematical literature. In chemistry, one finds several references on the Harary index [32-34] and several chapters in the book on topological indices and related descriptors in QSAR and QSPR [35-39]. According to Trinajstic and coworkers [32], the Harary index w based on the chemists intuitive expectation that distant sites in a structure should influence each other less than the near sites  

This observations on the Harary index and Wiener index oppose intuitive reasoning that the outer bonds, the more exposed bonds, should have greater weights than the inner bonds because the outer bonds are associated with the larger part of the molecular surface and are consequently expected to make a greater contribution to the physical and chemical properties. 

The square root distance matrix does reduce automatically the influence of more distant neighbors in a natural way, so it is of interest. In that respect, even better is the n-root matrix, illustrated at bottom in Table 8.3 on 3-methylhexane and norbomane because now with increases in distance the root exponent also increases, which more drastically decreases the role of vertices at larger distances. We are not proposing the square root matrix and the related n-root matrix as an answer that will cure the ill features of the distance matrix as a source for construction of molecular descriptors to be used in structure-property-activity studies, but more to illustrate an alternative modification of the distance matrix for construction of topological indices. 

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Some graph-theoretical matrices derived from the distance matrix are reported below. Generalized distance matrices, denoted as D, are derived from the distance matrix by raising... 

The distance distribution moments, denoted as D, are the moments of the distribution of topological distances dij in a molecular graph, derived from generalized distance matrices defined for positive integer X values [Klein and Gutman, 1999] ... 

The distance matrix A, which holds the relative distances (by whatever similarity measure) between the individual confonnations, is rarely informative by itself. For example, when sampling along a molecular dynamics trajectory, the A matrix can have a block diagonal form, indicating that the trajectory has moved from one conformational basin to another. Nonetheless, even in this case, the matrix in itself does not give reliable information about the size and shape of the respective basins. In general, the distance matrix requires further processing. 

Note that although the bounds on the distances satisfy the triangle inequalities, particular choices of distances between these bounds will in general violate them. Therefore, if all distances are chosen within their bounds independently of each other (the method that is used in most applications of distance geometry for NMR strucmre determination), the final distance matrix will contain many violations of the triangle inequalities. The main consequence is a very limited sampling of the conformational space of the embedded structures for very sparse data sets [48,50,51] despite the intrinsic randomness of the tech-... 

Equations (125) and (126) explicitly show that in the initial slope approximation the elements of the generalized mobility matrix can be expressed only in terms of integrals over the corresponding partial static structure factor. Both equations are valid as long as one assumes a Gaussian distance distribution of the distances r between the monomers i on arm a and monomers j on arm p. 

The classical definition for the graph centre is not helpful for cyclic graphs where often a large number of vertices (most of them topologically non-equivalent) appear as central. Recently, Bonchev, Balaban and Mekenyan 67) proposed a generalized concept for the graph centre, and several centric indices were derived on this basis. The new definition consists of four hierarchically ordered criteria based on the distance matrix 1) the smallest maximum distance in the row or column of the vertex ... 

This is a non-parametric method that calculates the distances matrix between all n observations and uses the following assignation rule sample is assigned to the group most represented among the nearest k observations . Generally k is odd, and the size of the groups is also taken into account. 

Distance matrix A matrix of distances, originally defined as from each a-carbon atom in a protein to every other a-carbon atom. The matrix is indexed in the consecutive order of the amino acid residues in the protein. The term is also used now more generally. 

Another generalization of the Balaban index J, so as to account for heteroatoms in the molecule, is the - Barysz index calculated on the - Barysz distance matrix. 

A generalization of the expanded distance matrix was proposed by Diudea [Diudea and Gutman, 1998] in order to define new matrices derived from the Hadamard matrix product between the distance matrix D and a general square A xA matrix M as ... 

A generalization of the Wiener index to account for heteroatoms and multiple bonds was proposed based on the Barysz distance matrix. 

Note. The vertex-distance-vertex-degree matrices with P = 0 were called v d matrices by Perdih [Perdih and Perdih, 2002a] and the general distance-degree matrix was denoted by G a,b,c) [Perdih and Perdih, 2004], whose elements are v vj dl [ Perdih and Perdih, 2003c], where v denotes the vertex degree 6 and a, b, and c are the parameters corresponding to p, y, and a, respectively. 

The diagonal elements of these matrices are equal to zero. Using the notations adopted in this book, the general distance-degree matrix elements are defined as the following ... 

Generalized Topological Distance Indices —> distance matrix... 

Examples of molecular descriptors derived from generalized matrices are distance distribution moments and W% indices from the distance matrix and molecular profiles from the geometry matrix. Moreover, vertex Zagreb matrix (X = 2) and modified vertex Zagreb matrix (X = —2) are a generalization of the vertex degree matrix, and the generalized... 

For vertex- and edge-weighted molecular graphs, the weighted distance matrix is generally defined as a square symmetric A x A matrix, A being the number of graph vertices. 

Derived from the Barysz distance matrix, a general weighting scheme was proposed by Ivanciuc [Ivanciuc, Ivanciuc et al, 1999a Ivanciuc, 2000a, 2000i] in terms of the conventional bond order n and any atomic property Pj. The vertex weight Wi associated with the vertex V was defined as... 

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