Bond adjacency 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. 

At the University of Santiago de Compostela, recent work has developed a family of 2D molecular descriptors based on the local spectral moments of a bond adjacency matrix [25b,26], Particular attention has been paid to using the bond spectral moments of the bond matrix that correspond to the central bond of a dihedral angle as descriptors for the dihedral angle. 

ACF = atom-centered fragment AI = artificial intelligence BAM = bond adjacency matrix CHEMICS = combined handling elucidation method for interpretable chemical structures COSY = correlated spectroscopy FBMX = free-bond connection matrix HMBC = heteronuclear multiple bond correlation spectroscopy HMQC = heteronuclear multiple quantum coherence correlation spectroscopy SESAMI = systematic elucidation of structure applying machine intelligence. 

The CISOC-SES system is also structure-reduction based and shares some similarities with the approach described above for COCOA. The procedure also uses a bond adjacency matrix - the free-bond connection matrix - to represent the bonding possibilities between non-hydrogen atoms. However, in contrast to COCOA, individual half-bonds of element groups are not specified in the matrix, only the number of equivalent free bonds, i.e., all possible hybridizations of an element group consistent with the number of free bonds are allowed. The program also utilizes a single-layered, spherical atom-centered fragment in structure reduction and can accept user-entered constraints. 

The function of CISOC-SES (see Section 4.3) parallels that of SESAMI, i.e., to directly reduce the molecular formula of an unknown to a small list of plausible alternatives using ID and 2D NMR spectral data. The process consists of the three successive stages. In the first, a bond adjacency matrix is created, the free-bond connection matrix (FBMX) in this case. Atom correlations derived from 2D NMR experiments are then extensively used to reduce the FBMX. In the final stage, reduction to a one-to-one mapping of bonds is achieved by means a simple, recursive, depth-first search procedure that yields all compatible structures. C NMR chemical shift data and simple heuristics are used to guide this process and increase its efficiency. 

The adjacency matrix of a molecule con.si.sting of n atom.s i.s a square (n / n) matrix. with the entric.s giving all the connectivities of the atoms. The intersection of a row and a column obtains a value of 1 if the corresponding atoms are connected. If there is no bond between the atoms being considered, the position in the matrix obtains the value 0. Thus, this matrix representation is a Boolean matrix with bits (0 or I) (Figure 2-13). 

With such a matrix representation, the storage space is dependent only on the number of nodc.s (atoms) and independent of the number of bonds. As Figure 2-14 dcmon.stratcs, all the e.sscntial information in an adjacency matrix can also be lound in the much smaller non-rediindant matrix. But the adjacency matrix is unsuitable for reconstructing the constitution of a molecule, because it does not provide any information about the bond orders. 

The bond matrix is related to the adjacency matrix but gives information also on the bond order of the connected atoms. Elements of the matrix obtain the value of 2 if there is a double bond between the atoms, c.g, between atoms 2 and 3... 

In Chapter 1 we have stated that the classical structural theory is the only way to "visualise" the synthesis of a more or less complex organic compound. However, all or most of the information given by a structural formula can also be expressed.by a matrix (see also Appendix A-1). There are different kinds of matrices for example, the adjacency matrix J, which originates in graph theory and indicates only which atoms are bonded, or the connectivity matrix C, whose off-diagonal entries are the formal covalent bond orders. For instance, the corresponding matrices of hydrogen cyanide are ... 

Matrix calculations are carried out for theoretical calculation of the unperturbed dimensions of e/ s-1,4-polybutadiene which takes into account the particular nature of the energy barriers to the rotation around single C—C bonds adjacent to the double bonds. 

The Pauling bond order (PBO) concept was introduced by L. Pauling, L. O. Brockway, and J. Y. Beach [1] (1935). The determination of the PBOs requires counting Kekule structures in GBSs. N. S. Ham [16] (1958) proved that, for a KBS, Pauling s bond order pu for neighbouring carbon atoms i, j equals the bond order defined by K. Ruedenberg [17] (1954) which results from LCAO-MO considerations further, he showed that ptJ is equal to the entry in position (i, j) of the inverse A 1 of the adjacency matrix A of the system (note that, by Eq. (2), det A = K2 > 0). 

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. 

We first mention an interesting result by Heilbronner [26] relating the inverse of the adjacency matrix with the Pauling bond order ... 

As you can see in Figure A2.2[a] the bond length of the Hiickel regular orbit cage is about 0.263 units and so the Adjacency matrix is constructed, Figure A2.2[b], over the 120 x 120 matrix using, in each cell of the matrix, the conditional formula, for example, in cell F 138... 

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] ... 

The -> adjacency matrix A of a molecular graph G is an example of binary sparse matrix, only the off-diagonal entries i-j, where v, and Vy are adjacent vertices, i.e. vertices connected by a bond, being equal to one. Using the adjacency matrix as multiplier in the Hadamard product it follows ... 

This is the simplest graph invariant obtained from the -> adjacency matrix A, defined as the number of bonds in the -> molecular graph (7 where multiple bonds are considered as single edges. Bond number is calculated as half the - total adjacency index Ay ... 

The determinant of the - adjacency matrix A. It was observed that this determinant is often equal to zero and this is a necessary and sufficient condition for the presence of non-bonding molecular orbitals in Hiickel theory. The actual numerical value of det A is correlated to the thermodynamic stability of the molecule [Graovac and Gutman, 1978 Trinajstic, 1992]. 

The determinant of (A" -i- G) was also proposed as a molecular topographic descriptor where A is the weighted adjacency matrix (- weighted matrices) where the entries corresponding to bonded atoms are - bond distances and G the geometry matrix [Mihalic et ai, 1992a]. 

The edge degree e, provides the simplest information related to the considered bond and is calculated from the edge adjacency matrix as follows ... 

Therefore, the ith edge degree calculated on the weighted edge adjacency matrix is the sum of the bond orders associated with all e, bonds adjacent to the / th edge ... 

In analogy with the bond order-weighted edge adjacency matrix, a resonance-weighted edge adjacency matrix E was also proposed, replacing the bond orders with parameters kc-x used in the Hiickel matrix and related to the resonance integral Pc-x of the bond between the heteroatom X and the carbon atom by the relationship ... 

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