Local vertex invariants

Ivanciuc, O., Balaban, T.S., and Balaban, A.T., Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices, J. Math. Chem., 12 309-318, 1993. 

Balaban S, Filip PA, Ivanciuc O, Computer generation of acyclic graphs based on local vertex invariants and topological indices derived canonical labelling and coding of trees and alkanes, J. Math. Chem., 11 79-105, 1992. 

Moreover, the adjacency matrix can be transformed into a decimal adjacency vector a of A elements each being a local vertex invariant obtained by the following expression [Schultz and Schultz, 1991] ... 

To obtain spatial autocorrelation molecular descriptors, function /(x,) is any physico-chemical property calculated for each atom of the molecule, such as atomic mass, polarizability, etc., and - local vertex invariants such as - vertex degree. Therefore, the molecule atoms represent the set of discrete points in space and the atomic property the function evaluated at those points. 

These are global molecular descriptors derived from an H-depleted molecular graph where each vertex is weighted by a local vertex invariant called Atom-in-Structure Invariant Index (ASII) defined as [Bangov, 1988] ... 

Some molecular descriptors and local vertex invariants proposed as a generalization or modification of the original connectivity indices are reported below. 

The maximum path sum of the / th vertex, denoted by MPVS, is a local vertex invariant defined as the sum of the length of the longest paths between vertex v, and any other vertex in the molecular graph, i.e. 

The maximum/minimum path sum of the i th vertex, denoted by MmPVS, is a local vertex invariant defined as the sum of the lengths of the longest and shortest paths between vertex v, and any other vertex in the molecular graph. It is calculated as the sum of elements over the / th row and / th column in the A/D matrix, or, alternatively, as the sum of the - vertex distance degree o, calculated on the distance matrix D and the maximum path sum MPVS, of the / th vertex calculated on the detour matrix A ... 

All these quantities are -> local vertex invariants. High values of the vertex distance sum o are observed for -> terminal vertices while low values for -> central vertices. Moreover, among the terminal vertices, the vertex distance degrees are small if the vertex is close to a branching site and larger if the terminal vertex is far away. 

Bond multiplicity is taken into account by augmenting the edge distance matrix with a supplementary column and row where the elements are conventional bond orders, therefore obtaining an edge distance matrix for multigraphs [Bonchev, 1983]. All the local vertex invariants and molecular descriptors defined above can also be calculated on this matrix. 

The first proposed electronegativity-based connectivity index was the DSI index [Diudea and Silaghi-Dumitrescu, 1989] based on the -> Sanderson group electronegativity ESG used as a local vertex invariant. It is defined as ... 

The electrotopological states are local vertex invariants. After rescaling, they are also used as atomic weighting factors for the calculation of the -> WHIM descriptors. 

Local vertex invariants proposed by Balaban [Balaban and Catana, 1993] with the aim of obtaining high discrimination among the vertices of an -> H-depleted molecular graph. They are denoted by c, and are expressed in logarithmic units as ... 

The extended adjacency matrices EA are - weighted adjacency matrices Ax.A whose elements are defined as a function of local vertex invariants of the -> adjacency... 

Other topological molecular descriptors can be obtained by using suitable functions applied to - local vertex invariants, the most common functions are atom and/or bond additives, resulting in descriptors which give correlation of physico-chemical properties that are atom and/or bond additives themselves. - Zagreb indices and -> ID numbers are derived according to this approach. 

These indices are usually calculated by assigning a weighting factor Wy to each - path or - walk of the - molecular graph with v, and Vy as endpoints. By summing all weighted paths (or walks) starting from the ith vertex, atomic ID numbers (AID) are obtained as -> local vertex invariants characterizing the atomic environment. 

Molecular descriptors calculated as - information content of molecules. Different criteria are used for defining - equivalence classes, i.e. equivalency of atoms in a molecule such as chemical identity, ways of bonding through space, molecular topology and symmetry, - local vertex invariants [Bonchev, 1983]. 

The weights w can be any chemical or topological atomic properties. Examples of chemical -> atomic properties are - van der Waals volume, atomic mass, - polarizability examples of -> local vertex invariants are - vertex degree, - path degree, - walk degree. 

By this matrix, the - information layer index is calculated. Moreover, four different local vertex invariants derived from the cardinality layer matrix have been proposed [Wang et al., 1994] as follows ... 

Derived from layer matrices, two main types of -> local vertex invariants were defined on the basis of two types of operators, the centric operator c, and the centro-complexity operator x, [Diudea, 1994 Diudea et al., 1995b] ... 

The sums of the local vertex invariants X, and c, over all of the atoms give the corresponding molecular descriptors, called centrocomplexity topological index X and centric topological index C, respectively ... 

Local vertex invariants are used to calculate several molecular - topological indices by applying different operators such as addition of LOVIs, addition of squares of LOVIs, addition of reciprocal geometric means for any pair of adjacent vertices. Moreover, they can be used to obtain - canonical numbering of molecular graphs and compare molecules in order to study - molecular branching and centricity. 

A set of local vertex invariants EFTI/ was derived from - fragment topological indices, when one non-hydrogen atom at a time is considered ... 

Another set of local vertex invariants was proposed by Ivanciuc [Ivanciuc, 1989] as combinations of topological distances d/y and vertex degrees 6 by the following general expression ... 

The atomic properties constitute the weights used to characterize molecule atoms the most common atomic properties are atomic mass, - atomic charge, -> van der Waals radius, -> atomic polarizability, and hydrophobic atomic constants. Atomic properties can also be defined by the - local vertex invariants (LOVIs) derived from graph therory. 

Local vertex invariants (LOVIs) obtained as the solutions of a linear equation system defined as ... 

Closely related to MPR descriptors are local vertex invariants called graph potentials denoted by U,- [Golender et al., 1981 Ivanciuc et ai, 1992]. They are calculated as the solutions of a linear equation system defined as ... 

Based on the length of the paths in the molecular graph, other local vertex invariants and molecular descriptors have been proposed. 

Randic-Razinger index -> local vertex invariants... 

Schultz-type indices based on eigenvectors were recently proposed [Medeleanu and Balaban, 1998] as - local vertex invariants and molecular descriptors, on the basis of the eigenvector of adjacency and distance matrices associated with the lowest (largest negative) eigenvalue. The LOVIs are derived from the following A-dimensional column vectors ... 

Applying the - row sum operator to the sequence matrices, local vertex invariants, atomic sequence count asci, are obtained as ... 

---