Hydrogen-suppressed graph

Bond indices /3 and /3 can be assigned to each edge of the hydrogen suppressed graph at the two vertices, / and... 

Finally, the atomic and bond indices can be combined to give indices for the whole unit. The zeroth-order connectivity indices and Oy for the entire molecule can be calculated as a summation over the vertices of the hydrogen suppressed graph, that is ... 

Oorb Order of neighborhood when ICr reaches its maximum value for the hydrogen-suppressed graph 7orb Information content or complexity of the hydrogen-suppressed graph at its maximum... 

Fig. 1. An example of two hydrogen-suppressed graphs G1 G2 and a common substructure CSIG,, G2) and the maximum common substructure MCS(G1 G2) are shown above. The Tanimoto similarity index and the distance between the two chemical graphs are computed below.

Propane Molecular graph Hydrogen-suppressed graph... 

Figure 6. The hydrogen-suppressed graph representations of allylic metal-cyclopentadienyl complex...

The values of two indices (8 and 8V, see Table 2.1 and the subsequent discussion), which describe the electronic environment and the bonding configuration of each non-hydrogen atom in the molecule, are next assigned, and listed at the vertices of the hydrogen-suppressed graph. 

Figure 2.2. Calculation of the zeroth-order and first-order connectivity indices, using vinyl fluoride as an example. Hydrogen-suppressed graph with (a) 8 values at the vertices and (1 values along the edges, and (b) 8V values at the vertices and Pv values along the edges, (c) Summation of the reciprocal square roots of the 8 values to calculate °%, of the 8V values to calculate °%v, of the (3 values to calculate and of the Pv values to calculate 1%v.

The number N of vertices in the hydrogen-suppressed graph, the zeroth-order connectivity indices and °%v, and the first-order connectivity indices l% and 1xv, are listed in Table 2.2 for a diverse set of 357 polymers. All of the common polymers are included in this table. Many specialized and/or exotic polymers are also included. The polymers are listed in the order of increasing N. The easiest way to look up the connectivity indices of a polymer in Table 2.2 is to draw the stmcture of the repeat unit of the polymer, count the number of non-hydrogen atoms in this repeat unit, and search among the polymers with that value of N. For example, poly(vinyl fluoride), which is shown in Figure 2.3, has N=3. Its connectivity indices can be... 

As mentioned in a footnote to Table 2.1, the use of 8v=l/3 or 8v=4/9 for silicon atoms, as obtained from the definition of 8V (Equation 2.1), causes the overestimation of the effect of the extra inner shell of electrons in silicon atoms on certain physical properties. Whenever this happens, the replacement Si—>C (i.e., 8V=3 or 4) will be made in calculating the valence connectivity indices to correlate that property. For such properties, the differences between Si and C atoms will be taken into account by introducing an atomic correction term for the number of silicon atoms in the repeat unit. The alternative sets of °%v and values obtained for silicon-containing polymers by making the replacement Si—>C in the hydrogen-suppressed graph of the polymeric repeat unit, are listed in Table 2.3. 

Table 2.3. Alternative °%v and 1%V values obtained by replacement Si—>C in hydrogen-suppressed graph of repeat unit and used in some correlations for silicon-containing polymers. 

As can be seen from equations 2.4-2.7, from Table 2.2, and most dramatically from figures 2.5-2.8, the % values are also extensive properties. They are sums over all vertices or edges of the hydrogen-suppressed graph. The number of terms in each summation increases in direct proportion to the size of the molecule or the polymeric repeat unit. This is the reason why the % values are proportional to N to a good approximation. They are, therefore, logical choices of topological descriptors to correlate with extensive properties. 

The set of linear regression coefficients (a or b), correction terms (if needed), and constant c in Equation 2.10, are adjustable parameters. The % and % values are determined exactly from the hydrogen-suppressed graph of the repeat unit. There is no additive constant term in Equation 2.9. A constant does not change as a function of the amount of material present, so that it is an intensive property which does not belong in correlations for extensive properties. For example, so long as there is some material present, the density (an intensive property) has the same constant positive value. On the other hand, the total volume becomes infinitesimally small (approaches zero) in the limit of an exceedingly small quantity of any material. The omission of the constant in Equation 2.9 is therefore essential to prevent the introduction of a computational artifact into correlations for extensive properties. 

In the last term in the definition of NyKH, the summation is over vertices of the hydrogen-suppressed graph with 5=1 linked to aromatic rings in the backbone of the repeat unit. Nrow is... 

The twelve structural parameters defined above are all extensive variables. In order to convert them into intensive variables for use in the correlation for Tg, which is an intensive property, they will all be scaled by the number N of vertices in the hydrogen-suppressed graph of the repeat unit, as described by Equation 2.8 in Section 2.C. In other words, xj/N, x2/N,. .., x12/N, will be used as linear regression variables in the correlation for Tg. 

Table 8.2. Experimental refractive index n at room temperature, number of rotational degrees of freedom Nrot and correction index Nref used in the correlation for n, and the fitted value of n, for 183 polymers. The number N of vertices in the hydrogen-suppressed graph of the polymeric repeat unit, and the connectivity indices °%, °%v and 1%v, all of which are also used in the correlation equation for n, are listed in Table 2.2. 

A detailed analysis showed that o depends strongly both on the nature of the connectivity and conformations of the chain backbone and on the relative size of the side group portion of the hydrogen-suppressed graph of the repeat unit. Since the dependence of a on these two types of structural factors is not described adequately by the parameters that were defined and used in earlier chapters, four specialized indices were defined and used in the correlation for o. 

Finally, a side group index SG was defined as the fourth power of the fraction (N - NBB)/N of the vertices of the hydrogen-suppressed graph of the repeat unit that are in its side groups ... 

The most important index was SG. It had a correlation coefficient of 0.8337 with o, so that there is a fairly strong correlation between o and increasing side group size as SG is the, fourth power of the fraction of the vertices of the hydrogen-suppressed graph that are in side groups. 

The correction index N, containing both atomic and group correction terms, was defined by Equation 12.30, and used to obtain an improved correlation (Equation 12.31) with a standard deviation of only 0.7 and a correlation coefficient of 0.9990 for J. In Equation 12.31, N is the number of vertices in the hydrogen-suppressed graph. 

The thermal conductivity is an intensive property (Section 2.C). In other words, its value is independent of the size of the system being considered. Consequently, the intensive ( -type) connectivity indices, which are defined by Equation 2.8 as the corresponding extensive (X type) connectivity indices divided by the number N of vertices in the hydrogen-suppressed graph (i.e., =%/N), were used in the correlation for X(298K). 

Figure 17.5. Hydrogen-suppressed graph of the a-naphthyl group in (a) poly(a-vinyl naphthalene) and (b) poly(a-naphthyl methacrylate). The simple atomic index 8 (see Chapter 2) is shown at the vertices. The products of pairs of 8 values are shown along the edges. The two graphs differ slightly because the vertex outside the box has 8=3 in (a) and 8=2 in (b), resulting in a small difference between the contributions of the a-naphthyl unit to the first-order connectivity index Both graphs make the same contribution to the zeroth-order index °x-...

Figure 17.6. Hydrogen-suppressed graph of polyoxynaphthoate, with the value of the simple atomic index 5 indicated at selected vertices. The lack of periodicity at the boundaries of the box, which contains a subunit smaller than the full repeat unit, causes different values to be calculated for the contribution of the group enclosed in the box to 1 % if only one of the two bonds linking it to the rest of the repeat unit is considered. One of these bonds contributes (1/60.5) o.4082, and the other bond contributes (l/9°-5)=0.3333. The correct contribution of these two bonds to is the average of these two numbers, i.e., 0.5-[(l/60-5) + (l/9°-5)]=O.3708. 

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