Molecular graphs vertex-weighted

For vertex-weighted molecular graphs, a weighted modified Wiener index was defined as [Gutman and Zerovnik, 2002]... 

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. 

From vertex-weighted molecular graphs, edge weights can be derived from some combination of the weights of the two vertices incident to the edge ... 

The adjacency matrix A(G) of a molecular graph G with N vertices is the square NxN symmetric matrix in which [A],j=l if vertex Vj is adjacent to vertex Vj and [A],j=0 otherwise. The adjacency matrix is symmetric, with all elements on the main diagonal equal to zero. The sum of entries over row i or column i in A(G) is the degree of vertex Vj, 5j. Usually, the adjacency matrix is based on weighted molecular graphs in which heteroatoms are represented as vertex parameters and... 

Ivanduc, 0., Ivanciuc, T., Balahan, A. T. Vertex- and edge-weighted molecular graphs and derived structural descriptors. In Topological Indices and Related Descriptors in QSAR and QSPR, Devillers, J. Balahan, A. T. (eds.), Gordon ... 

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] ... 

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. 

A layer matrix LM of a - molecular graph (7 is a rectangular unsymmetric matrix Ax(D + 1), A being the number of molecule atoms and D the topological diameter. The entry i-k (Imu ) is the sum of the weights of the vertices located in the concentric shell (layer) at topological distance k around the vertex v, [Diudea et al., 1991 Diu-dea, 1994 Skorobogatov and Dobrynin, 1988]. The kth layer of the vertex v, is the set of vertices defined as follows ... 

The general formula for the calculation of connectivity-like indices, which uses the row sums VSj of a —> graph-theoretical matrix as the local vertex invariants, was called by Ivanciuc Chi operator [Ivanciuc, Ivanciuc et al, 1997 Ivanciuc, 2001c], Specifically, for any square symmetric (Ax A) matrix M(iv) representing a molecular graph with Avertices and a —> weighting scheme w, the Chi operator is defined as... 

These are molecular descriptors calculated from the H-depleted molecular graph by analogy with the —> hyper-Wiener index. The general formula, called by Ivanciuc hyper-Wiener operator [Ivanciuc, Ivanciuc et al., 1997 Ivanciuc, 2001c], to calculate hyper-Wiener-type indices from vertex- and/or edge-weighted molecular graphs is... 

Moreover, matrices can be derived from unweighted or vertex- and/or edge-weighted molecular graphs in the latter case, several weighted matrices can be obtained depending on the —> weighting scheme. 

An example of a reduced graph is a molecular graph composed of weighted edges and with vertex number equal to the number of functional groups perceived by using predefined rules [Takahashi, Sukekawa et al., 1992]. 

The Szeged property matrices SZyP are square unsymmetrical Ax A weighted matrices derived from a vertex-weighted molecular graph. Each off-diagonal entry i-j is a function of a selected property of vertices P (,j) lying closer to the focused vertex V [Minailiuc, Katona et al,... 

By this approach, a family of fiexible molecular descriptors based on different mathematical functions obtained from the molecular graph <5 is defined. The basic idea is to vary the correlation weights CW ofthe different atom-types under consideration, aimed at obtaining an as high as possible correlation coefficient between experimental and calculated values of a selected molecular property. Atom-types are defined using chemical information and local vertex invariants. The general form of the molecular descriptor D in terms of a number of selected atom-types is... 

Derived from vertex-weighted molecular graphs, the augmented vertex degree matrix D (6) is an unsymmetrical weighted distance Ax A matrix based on the concept of —> augmented valence and defined as [Randic, 2001b Janezic, Milicevic et al., 2007] ... 

Based on these weighting schemes, several —> Ivanciuc weighted distance matrices were proposed to represent vertex- and edge-weighted molecular graphs [Ivanciuc, 2000i. ... 

Ivanciuc, O. (2000a) Design of topological indices. Part 12. Parameters for vertex- and edge-weighted molecular graphs. Rev. Roum. Chim., 45, 289-301. 

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