Molecular eccentricity

Among the - eigenvalue-based descriptors, molecular eccentricity is a shape descriptor obtained from the eigenvalues of the -> inertia matrix I defined as [Arteca, 1991] ... 

Let us now indicate the principal moments of inertia of a molecule (i.e., the eigenvalues of the matrix I) by X 1 s / < 3. The anisometry of the molecule can be expressed in terms of these moments. We can define a simple descriptor in analogy with the eccentricity of planar ellipses. The ellipse s eccentricity is the ratio (M — rn y lM, where M and m, respectively, are the length of the major and minor elliptical axes." Similarly, we can define a molecular eccentricity e in terms of the principal moments of inertia as ... 

Anisometry. Characterized by properties depending on inertia axes, symmetry axes, and distance distributions. Descriptors include the moments of inertia X,, the molecular eccentricity e, the asphericity fl, and the length of the principal axes of minimal ellipsoidal cavities, as well as distance matrices, contact maps, and some graph-theoretical indices. 

In the second oxidation method, a metalloporphyrin was used to catalyze the carotenoid oxidation by molecular oxygen. Our focus was on the experimental modeling of the eccentric cleavage of carotenoids. We used ruthenium porphyrins as models of cytochrome P450 enzymes for the oxidation studies on lycopene and P-carotene. Ruthenium tetraphenylporphyrin catalyzed lycopene oxidation by molecular oxygen, producing (Z)-isomers, epoxides, apo-lycopenals, and apo-lycopenones. 

An important theoretical development for the outer-sphere relaxation was proposed in the 1970s by Hwang and Freed (138). The authors corrected some earlier mistakes in the treatment of the boundary conditions in the diffusion equation and allowed for the role of intermolecular forces, as reflected in the IS radial distribution function, g(r). Ayant et al. (139) proposed, independently, a very similar model incorporating the effects of molecular interactions. The same group has also dealt with the effects of spin eccentricity or translation-rotation coupling (140). 

Molecular Phenotypes of Concentric and Eccentric Cardiac Hypertrophy... 

MOLCONN-Z EduSoft, LC www.eslc.vabiotech.com/molconn/ Molecular connectivity, molecular connectivity difference, and kappa shape indices, E-state indices, atom-type and group-type E-state indices, topological equivalence classification of atoms, other topological indices, counts of subgraphs paths, rings, clusters, etc. vertex eccentricities... 

From the eccentricity definition, a graph Q can be immediately characterized by two - molecular descriptors known as topological radius R and topological diameter D. The topological radius of a molecule is defined as the minimum atom eccentricity and the topological diameter is defined as the maximum atom eccentricity, according to the following... 

Other simple molecular descriptors are calculated as functions of atom eccentricity [Konstantinova, 1996], viz. eccentricity q, average atom eccentricity f], and eccentric Aq are defined, respectively, as the following ... 

A molecular descriptor based on both atom eccentricity t] (calculated from the -> distance matrix D) and - vertex degree 8 (calculated from the -> adjacency matrix A) in the H-depleted molecular graph [Sharma et al., 1997]. It is defined as the sum of the products between eccentricity and vertex degree 8 over the A atoms of the graph ... 

Molconn-Z Molconn-Z (http //www.edusoft-lc.com/molconn/) is commercial software that calculates molecular connectivity, shape, and information indexes such as molecular connectivity chi indexes, kappa shape indexes, electrotopo-logical state indexes, molecular connectivity difference chi indexes, topological indexes, counts of subgraphs, and vertex eccentricities [82], The software is available for multiple platforms, which include UNIX/LINUX, Windows, and Mac OS-X. 

Detour matrix A for the H-depleted molecular graph of ethylbenzene. is the atom detour eccentricity, MPVSj is the maximum path sum of the ith vertex, and w is the detour index. 

Other molecular descriptors based on atom eccentricity values combined with other —> local vertex invariants are the eccentric connectivity index, the —> eccentric distance sum, the connective eccentricity index, the eccentric adjacency topochemical indices, the superadjacency index, and the —> eccentricity-hased Madan indices. 

These are a series of graph invariants calculated on the H-depleted molecular graph and defined as different combinations of —> atom eccentricity T], which is the maximum topological distance from an atom, and other LOcal Vertex Invariants. Eccentricity-based Madan indices are listed in Table El where index names, formulas, and bibliographic references are given. 

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