Molecular graph matrices

Ivanciuc, O., Ivanciuc, T and Diudea, M.V. (1997a). Molecular Graph Matrices and Derived Structural Descriptors. SAR QSAR Environ.Res., 7,63-87. 

O. Ivanciuc, T. Ivanciuc, and M.V. Diudea, Molecular graph matrices and derived structural... 

By definition, the Q coefficient is equal to 1 for any pair of adjacent vertices (k = 1), regardless of the parameter values. Note that the coefficients Q are the elements of the so-called generalized molecular-graph matrix F, which is a square symmetric Ax A matrix, defined as... 

It can be noted that function i was used by Ivanciuc to define the —> distance-valency matrices, function was used by Kier-Hall to define the —> electrotopological state indices, function y was proposed by Randic to calculate the —> alimented valence of vertices, and function y was proposed by Estrada in the generalized molecular-graph matrix, where x is a user-defined parameter or a parameter to be optimized during the searching for the best QSAR correlations. 

Central the molecular graph is completely coded (each atom and bond is represented) matrix algebra can be used the niimber of entries in the matrix grows with the square of the number of atoms in ) no stereochemistry included... 

A major disadvantage of a matrix representation for a molecular graph is that the number of entries increases with the square of the number of atoms in the molecule. What is needed is a representation of a molecular graph where the number of entries increases only as a linear function of the number of atoms in the molecule. Such a representation can be obtained by listing, in tabular form only the atoms and the bonds of a molecular structure. In this case, the indices of the row and column of a matrix entry can be used for identifying an entry. In essence, one has to distinguish each atom and each bond in a molecule. This is achieved by a list of the atoms and a list of the bonds giving the coimections between the atoms. Such a representation is called a connection table (CT). 

The Wiener index was originally defined only for acyclic graphs and was initially called the path number [6]. "The path number, W, is defined as the sum of the distances between any two carbon atoms in the molecule in terms of carbon-carbon bonds". Hosoya extended the Wiener index and defined it as the half-sum of the off diagonal elements of a distance matrix D in the hydrogen-depleted molecular graph of Eq, (15), where dij is an element of the distance matrix D and gives the shortest path between atoms i and j. 

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... 

The basic mathematical structure which maps a certain molecule (molecular graph G) is the adjacency matrix of the molecular graph A(G). For a molecule having N atoms, A(G) is a square N x N matrix. Its entries ay have only two different values 1, and 0, due to the fact that two atoms in a molecule are in binary relation, being either connected or not connected ... 

Thangavel, P and P. Venuvanalingam, Algorithm for the Computation of Molecular Distance Matrix and Distance Polynomial of Chemical Graphs on Parallel Computers. J. Chem. Inf. Comput. Sci., 1993 33, 412—414. 

Much of the material we present in this paragraph is discussed exhaustively in two classical textbooks on the chemical graph theory [7, 8] and therefore we review it only very briefly. Let A(G) be the adjacency matrix pertinent to a molecular graph... 

The derivation of the topological distance matrix from the molecular graph is followed by the assignment of PPPs to the nodes of the graph. The following list provides chemical definitions of the five PPP types that are implemented in the CATS descriptor. The upper-case letter in parentheses is the abbreviation of each PPP type. Additionally, a functional group description is paired with its corresponding SMARTS in square brackets ... 

Fig. 3.2 Schematic of the CATS descriptor calculation, (a) The hydrogen-depleted two-dimensional molecular graph provides the input, (b) The graph is simplified for the distance matrix computation different bond orders are not considered (unweighted graph) and all element types are disregarded. The algorithm starts at an arbitrary chosen atom and visits all nodes of the graph in a breadth-first manner, thereby building up the distance matrix. The numbers at the vertices are used to reference individual atoms in the distance matrix.

Figure 6.13 The DIOP ligand a structure b molecular graph c adjacency matrix. The broken linesindicatetheshortestand longest P-P connectivity paths, Dpi p2 and AP1 P2, respectively. The adjacency matrix of a molecular graph is a matrix with rows and columns labeled by graph vertices v (i.e., the atoms), with a 1 or 0 in position (vp Vj) according to whether Vj or Vj are adjacent or not.

Based on the matrix expression of molecular graphs, we can calculate the length of paths connecting any pair of atoms, i.e., a series of consecutive edges that connect two... 

The spin-Hamiltonian VB theory is a very simple and easy-to-use semiempi-rical tool that is based on the molecular graph. It is consistent with the VB theory described in Chapter 3, albeit with some simplifying assumptions and a more limited domain of application. Typically, this theory deals with the neutral ground or excited states of conjugated molecules or other homonuclear assemblies with one electron per site. For large systems, it reproduces the results of PPP full Cl, while dealing with a much smaller Hamiltonian matrix. 

A numerical value associated with chemical constitution that can be used to correlate chemical structure with various physical properties, chemical reactivity, or biological reactivity. The numerical basis tor topological indices is provided (depending on how a molecular graph is converted into a numerical value) by either the adjacency matrix or the topological distance matrix. In the latter, the topological distance between two vertices is the number of edges in the shortest path between these. 

Another application of basic subgraphs arises from the possibility of relating the invariants of molecular graphs to the occurrence numbers of some basic subgraphs. Estrada has developed this methodology for spectral moments of the edge-adjacency matrix of molecular graphs - defined as the traces of the different powers of such matrix ... 

Thus, any spectral moment and hence the activities/properties of chemical compounds can be represented by contributions of corresponding fragments. This approach was further extended to molecular graphs containing heteroatoms by weighting the diagonal elements of the bond adjacency matrix. 

Most conventional way to define molecular graph G is to determine the adjacency matrix A G) = atjH,i,j =, where Oy = 1 if... 

FIGURE 2 Mathematical representation of chemical structure as (a) a graph (b) a molecular graph (c) an adjacency matrix. 

Representations of a graph, molecular graph, and their adjacency matrix are shown in Figs. 2(a), (b), and (c), respectively. It is apparent that the elements a, may be treated, again, as Boolean truth values of classical... 

Derived from the -> molecular graph <5, the adjacency matrix A represents the whole set of connections between adjacent pairs of atoms [Trinajstic, 1992]. The entries Oy of the matrix equal one if vertices v, and Vy are adjacent (i.e. the atoms / and j are bonded) and zero otherwise. The adjacency matrix is symmetric with dimension A x A, where A is the number of atoms and it is usually derived from an -> H-depleted molecular graph. 

The total adjacency index Ay is the sum of all the entries of the adjacency matrix of a molecular graph, and is twice the - bond number B [Harary, 1969a] ... 

For any square symmetric A x A matrix M representing a molecular graph with A vertices, the hyper-Wiener operator 3-Cy"W is defined as [Ivanciuc et al, 1997a] ... 

---