Atomic graph

Fig. 12.26 First- and second-order extreme shapes for four-, five- and six-atom graphs (the linear molecule gives rise to the minimum in each case).

Fig. 7.4. Representations of the Laplacian distributions of methane and methylfluoride. The figures in (a) are displays of the zero envelope of V p(r), those in (b) of the atomic graphs. The envelope encompassing the inner shell charge concentration on carbon appears as a small sphere. The envelopes of the bonded maxima in the VSCC of carbon also encompass the protons in CH and CH3F. There is a transfer of charge from C to F in CHjF and the bonded maximum along the C F axis is reduced to the small region lying between the C nucleus and the envelope on F. An atomic graph displays the connectivity of the critical points in a VSCC. The carbon nucleus is denoted by a solid cross, the positions of the remaining nuclei by open crosses. There is a bonded maximum, a (3, — 3) critical point in — V p, at each of the four vertices.

Fifjure 6. Angular and kinetic energy distribution of the outgoing hydrogen atoms. Graphs a) and c) corresponds to the H + I exit channel, while b) and d) ndth the H + I channel. In the case a) and b) the dopant is placed in the second shell, while in the case c) and d) the dopant is on the surface, c) Simulation of the phorodissociation of the dopant on the surface (histogram) compared with a pick-up photodissociation experiment (line with error bars), f) Simulation of the photodissociation of the dopant in the sub-surface shell (histogram) compared with a pick-up photodissociation experiment (lino with error bars). 

Figure 6 From atom graphs, to bond graphs, to correspondence graph.

Aray et a/. analysed the topology V p of bulk 3d transition metals and zoomed in on the atomic graphs. The authors were able to correlate the value of at the local minima with the experimental heats of adsorption for the O2 and CO molecules on the 3d transition metals. 

The atomic graph for Cl in the pseudo-octahedral geometry of CIF5 is illustrate in Figure 14. It is an irregular octahedron, the positions of the six vertices corresponding to the presence... 

The following graph shows the variation in atomic radius with increasing atomic number ... 

Figure li. . Graph oj h.p. against atomic number jor the Jirst transition series... 

Chapter 6). We note, however, that there is not a smooth increase in the magnitude of these properties as the atomic number increases the metals seem to divide into two sets, Sc-Mn and Mn-Zn with peaks at Ti-V and Co-Ni. and this is well illustrated by a graph of boiling point against atomic number (Figure 13.1). 

The use of larger particles in the cyclotron, for example carbon, nitrogen or oxygen ions, enabled elements of several units of atomic number beyond uranium to be synthesised. Einsteinium and fermium were obtained by this method and separated by ion-exchange. and indeed first identified by the appearance of their concentration peaks on the elution graph at the places expected for atomic numbers 99 and 100. The concentrations available when this was done were measured not in gcm but in atoms cm. The same elements became available in greater quantity when the first hydrogen bomb was exploded, when they were found in the fission products. Element 101, mendelevium, was made by a-particle bombardment of einsteinium, and nobelium (102) by fusion of curium and the carbon-13 isotope. 

Figure 2-11. Phenylalanine can be represented in graph theory as a labeled, weighted graph with different atom and bond types (as on the left-hand side).

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 minimum number of cycles is given by the nullity or Frerejacque number ( ) according to Eq. (5). It is the difference between the number of nodes a = atoms) and the number of edges h = bonds). The value of 1 stands for the number of compounds considered (here, one compound). This minimum number corresponds to the munber of chords. These are defined as nodes that turn a cyclic graph or structure into an acyclic one. 

Clearly, the next step is the handling of a molecule as a real object with a spatial extension in 3D space. Quite often this is also a mandatory step, because in most cases the 3D structure of a molecule is closely related to a large variety of physical, chemical, and biological properties. In addition, the fundamental importance of an unambiguous definition of stereochemistry becomes obvious, if the 3D structure of a molecule needs to be derived from its chemical graph. The moleofles of stereoisomeric compounds differ in their spatial features and often exhibit quite different properties. Therefore, stereochemical information should always be taken into ac-count if chiral atom centers are present in a chemical structure. 

Figure 6-2. Mappings between the que7 graph (Cq) and the target graph (Cr). Notation such as (2, 1, 3, 4) means that atom 2 of the que subgraph Gq is mapped to atom 1 from the target graph Cp...

In the worst case, the backtracking algorithm will form a search tree of depth n, where n is the number of atoms in the query graph. Also, in this case a separate sub-tree search process for each atom of the target graph will be initiated. That is why the linear multiplier m is apphed to Eq. (7). 

Table 6.1. Application of partitioning approach for substructure search optimization. According to their local properties, the atoms of the graphs in Figure 6-2 are separated into several classes.

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