Vertex degree

The Randic connectivity index, X, is also called the connectivity index or branching index, and is defined by Eq. (18) [7], where b runs over the bonds i-j of the molecule, and and dj are the vertex degrees of the atoms incident with the considered bond. 

Variance, 269 of a distribution, 120 significance of, 123 of a Poisson distribution, 122 Variational equations of dynamical systems, 344 of singular points, 344 of systems with n variables, 345 Vector norm, 53 Vector operators, 394 Vector relations in particle collisions, 8 Vectors, characteristic, 67 Vertex, degree of, 258 Vertex, isolated, 256 Vidale, M. L., 265 Villars, P.,488 Von Neumann, J., 424 Von Neumann projection operators, 461... 

Schultz MTI by valence vertex degrees Gutman Molecular Topological Index Gutman MTI by valence vertex degrees Xu index... 

ASVy Triplet index from adjacency matrix, distance sum, and vertex degree operation y — 1-5 DSVy Triplet index from distance matrix, distance sum, and vertex degree operation y — 1-5... 

ANVy Triplet index from adjacency matrix, graph order, and vertex degree operation y — 1-5... 

Flexible optimal descriptors have been defined as specific modifications of adjacency matrix, by means of utilization of nonzero diagonal elements (Randic and Basak, 1999, 2001 Randic and Pompe, 2001a, b). These nonzero values of matrix elements change vertex degrees and consequently the values of molecular descriptors. As a rule, these modifications are aimed to change topological indices. The values of these diagonal elements must provide minimum standard error of estimation for predictive model (that is based on the flexible descriptor) of property/activity of interest. 

The most basic element in the molecular structure is the existence of a connection or a chemical bond between a pair of adjacent atoms. The whole set of connections can be represented in a matrix form called the connectivity matrix [249-253]. Once all the information is written in the matrix form, relevant information can be extracted. The number of connected atoms to a skeletal atom in a molecule, called the vertex degree or valence, is equal to the number of a bonds involving that atom, after hydrogen bonds have been suppressed. 

A further extension of this approach was done by Kier and Hall8> so as to provide different values of the connectivity index for molecules depicted by one and the same graph, but differing by the chemical nature of atoms as well as by the presence of single, double or triple bonds. The valency of the atom i (vertex degree), Vj, is replaced by the atom connectivity ... 

Some simple relationships are also found among N2 (or, F), the vertex degree, Vj, the index Mb and the total adjacency A ... 

The code can be written more concise as an ordered sequence, omitting the vertex degrees since they are uniquely defined ... 

We list new the ( a,b, )-spheres that are 2-connected but are not polyhedra. Looking back at the proof of Theorem 9.1.1, we can notice that the hypothesis of 3-connectedness is used only for excluding a = 2 and, for k = 3, the case a = 3, i = 1. We consider those cases here, which, in particular, implies that the vertex degree k can be higher than 5. 

Further, A is used to generate the valence vector, v [11]. This vector indicates the vertex degree for each atom in G ... 

Diudea, M. V., O. Minailiuc, and A. T. Balaban, Regressive Vertex Degree (New Graph Invariants) and Derived Topological Indices. J. Comput. Chem., 1991 12, 527-535. 

The Zagreb group was the first to propose indices (Mj and M2) that were based directly on the graph adjacency matrix (Hall and Kier, 2001). Mj and M2 are defined as the sum of the squared vertex degrees (i.e., the number of edges with which it is connected, a,), and the sum of vertex degrees products (a ) over all pairs of adjacent vertices, respectively (Gutman et al., 1975) ... 

In Equation 5.9, the sum in the brackets equals the vertex degree products for half of the adjacency matrix. It is multiplied by two in order to obtain summation over all pairs of adjacent vertices. Note that by definition M2 is not equal to 21 IM,. Although it is difficult to derive bond contributions for the index based only on the vertex degrees (M,), the formal bond distribution of the Zagreb index M2 (Figure 5.9) shows that the terminal bonds are again underestimated, although by a different amount. 

The direct summation of vertex degree products in M2 has been changed in (G) to a summation of inverse-square-root terms. This specific function selection has been made to provide better correlations of with the properties of isomeric alkanes. This shows the high sensitivity of the new molecular descriptor to variations in molecular structure. More recently, some restrictions in the applicability of the inverse-square-root function to compounds with large numbers of atoms (Gutman and Lepovic, 2001) and new values for the exponent were investigated (Gutman and Lepovic, 2001 Estrada, 2002). 

Table 2 Equivalence classes- Sequences to the left are vertex-degrees listed in a descending order [lO]-...

All the above trees have vertex degrees of (4,3,2,l,l,l,l,l)> One might speak of a set of trees belonging to a given Young diagram, thus one may write ... 

The ith row sum of the adjacency matrix is called - vertex degree 6 defined as ... 

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. 

The idea behind these LOVIs is that usually the vertices with the highest distance sums have the lowest vertex degrees, thus enhancing the intramolecular differences [Baiaban, 1994a]. 

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