J. Chem. Inf. Comput. Sci. 2001, 41, 867-877
867
Topological Indices Based on Vertex, Distance, and Ring: On the Boiling Points of Paraffins and Cycloalkanes Chenzhong Cao* and Hua Yuan Department of Chemistry, Xiangtan Normal University, Xiangtan 411201, People’s Republic of China Received December 9, 2000
Vertex, distance, and ring (in cyclic compounds) are three essential structure elements of a molecular graph, based on which three new topological indices VDI, OEI, and RDI are proposed. Multiple regression analysis was carried out against the boiling points of 343 hydrocarbons(160 paraffins and 183 cycloalkanes) with VDI, OEI, RDI, and N2/3(N is the number of carbon atoms) together, and a good QSPR model was obtained: ln(577 - Bp) ) 6.729609 - 0.154107N2/3 + 2.285632 × 10-2VDI - 7.921410 × 10-3OEI 1.821962 × 10-2RDI (F ) 6455.09, r ) 0.9935, rms ) 6.44 °C, n ) 343). 1. INTRODUCTION
As we know, the physicochemical and biochemical properties of compounds depend strongly on their molecular structures. For decades, people have been trying to quantify molecular structures of compounds on the basis of graph theory and relate it to their properties. As yet, much more than 100 topological indices, such as Randic index1, Balaban index,2 and Schultz index,3 have been developed since Wiener reported the first well-known graph-theoretical descriptor, Wiener index.4 Most of these indices were derived from distance matrix or vertex adjacency as well as edge adjacency matrices. But a molecular graph includes more structure elements than both distance and vertex. So a more overall consideration should be taken into molecular structure while defining topological indices. In the earlier time, chemists tended to relate the properties of compounds with single structure descriptor.5 Though its calculation is simple, the obtained result is not satisfactory because only one descriptor is not enough to express all the molecular structure information. To remedy this defect, chemists now are inclined to regress more structure descriptors against the properties of compounds, by which better correlation can be expected despite of the more complicated operation. For example, in the NBP (normal boiling point)structure studies of alkanes, Liu6 employed 10 structure descriptors and obtained an equation with high correlation coefficient and low standard deviation. For another example, Estrada7 related the boiling points of cycloalkanes to molecular structures with six variables and got a good correlation result. It seems that the more variables that are employed, the better the result that can be attained. In fact, it is not the case. For instance, Cao8 regressed only four variables against the boiling points of alkanes and alcohols and still obtained the approving result (n ) 328, r ) 0.9990, s ) 6.021 °C). It implies that fewer descriptors may also get quite good QSPR results if the defined descriptors are elaborate. In general, a good QSPR/QSAR equation should require a high correlation coefficient, low standard deviation with * Corresponding author fax: 0086-0732-8291001; e-mail: czcao@ xtnu.edu.cn.
parameters as few as possible. But how many parameters are acceptable and what are they? For a long time, chemists have been concentrating on solving this problem. However it is not easy to give a satisfactory answer. Let us take a look into the molecular graph: the vertex, distance, and ring (in cyclic compounds) are three essential structure elements which can characterize most of the molecular structure information. The variables derived from the above three structure elements will improve the correlation between the properties of compounds and their molecular structures. In this paper we presented three new topological indices, OEI (odd-even index), VDI (vertex degree-distance index), and RDI (ring degree-distance index), and then carried out multiple regression analysis with these indices (plus carbon atom number N) against the boiling points of paraffins and cycloalkanes (including polycyclic and spirocycloalkanes with carbon atoms up to 10). A good result was obtained. 2. CALCULATIONS
In the hydrogen-suppressed graph, distance, Dij, is the length of the shortest path between vertex i and j; vertex degree, Vi, is the number of vertices adjacent to vertex i; and ring degree, Ri, is the minimum number of vertices (adjacent to the ith vertex) that must be removed to transform the ith vertex into an acyclic one. Each of them are represented by the corresponding matrix, i.e., distance matrix(N × N)D, vertex degree vector(1 × N)V, and ring degree vector(1 × N)R, respectively. Based on D, V, and R, we defined three new topological indices. (1) Odd-even index (OEI) N
N
OEI ) ∑ ∑[(-1)Dij-1S]
(1)
i)1 j*i
where N is the number of vertices in molecular graph. S is the derivative matrix from distance matrix D, whose elements are the squares of the reciprocal distances (Dij)-2, i.e., S ) [1/Dij2] (when i ) j, let 1/Dij2 ) 0). It means that the interaction between vertex i and j is in proportional to (Dij)-2. However, the behavior of odd distance is different from that
10.1021/ci000467t CCC: $20.00 © 2001 American Chemical Society Published on Web 07/03/2001
868 J. Chem. Inf. Comput. Sci., Vol. 41, No. 4, 2001
CAO
of even distance between vertex i and j, thus a coefficient (-1)Dij-1 is introduced into the expression so as to obtain the positive and negative values for odd distance and even distance Dij item, respectively. (2) Vertex degree-distance index (VDI): The interaction of vertex i against j is determined not only by the distance between i and j, but also by their vertex degrees. So we defined VDI as N
VDI ) (∏fi)1/N
(2)
VS ) [f1, f2, ‚‚‚, fN]
YUAN
Figure 1. Structure of 1-methylbicyclo[1.1.0]butane. Table 1. Interrelations of N2/3, OEI, RDI, and VDI N2/3
OEI
RDI
VDI
1.0000 0.8800 0.3452 4.3218 × 10-2
1.0000 6.7246 × 10-3 0.2578
1.0000 0.8095
1.0000
i)1
fi is the elements of vector(1 × N) VS obtained by V-multiply-S
AND
N2/3 OEI RDI VDI
(3) Table 2. Regression Results with Different Variables
(3) Ring degree-distance index (RDI): Because of the rigidity of the ring, the freedom of vertex in ring is smaller than that in the chain. Thus, another index RDI was proposed to distinguish the cyclic and acyclic hydrocarbons. N
RDI ) (∏gi)1/N
gi is the elements of vector (1 × N) RS obtained by R-multiply-S:
RS ) [g1, g2, ‚‚‚, gN]
(5)
It should be pointed out that we employ eqs 2 and 3 other N N than expressions (∑i)1 fi)/N and (∑i)1 gi)/N to calculate VDI and RDI because the values obtained from the former are more distinguishable for isomers than that from the latter. We take 1-methylbicyclo[1.1.0]butane for example to compute the indices OEI, VDI, and RDI. Figure 1 is the hydrogen-suppressed graph of 1-methylbicyclo[1.1.0]butane (where the digits are the random numberings of each vertex). Its calculation steps are as follows: (a) The distance matrix D and its derivative matrices
[
[ [
0 1 S ) 0.25 1 0.25
1 0 1 1 2 1 0 1 1 0.25
0 1 (-1)Dij-1 S ) -0.25 1 -0.25
2 1 0 1 2
1 1 1 0 1
0.25 1 0 1 0.25
1 0 1 1 -0.25
2 2 2 1 0
rms
N2/3 VDI OEI RDI
Single Variable 6871.39 0.95 2222.96 22.88
0.9761 0.0527 0.9311 0.2507
11.39 51.74 19.79 50.57
N2/3, VDI N2/3, RDI N2/3, OEI VDI, RDI VDI, OEI RDI, OEI
Two Variables 3434.06 4205.78 6860.51 57.44 1611.90 2369.34
0.9761 0.9804 0.9878 0.5025 0.9511 0.9659
11.40 10.74 8.75 42.73 17.40 14.29
N2/3, VDI, RDI N2/3, VDI, OEI N2/3, RDI, OEI VDI, RDI, OEI
Three Variables 6485.97 5646.91 4635.35 1612.63
0.9914 0.9901 0.9880 0.9667
7.32 7.73 8.61 14.02
N2/3, VDI, RDI, OEI
Four Variables 6455.09
0.9935
6.44
]
(c) The ring-degree vector R
R ) [1, 2, 1, 2, 0] (d) The V-multiply-S vector VS
(e) The R-multiply-S vector RS
0.25 0.25 0.25 1 0
-0.25 1 0 1 -0.25
1 1 1 0 1
V ) [2, 3, 2, 4, 1]
r
VS ) [7.75, 8.25, 7.75, 8, 5.75]
1 1 1 0 1
(b) The vertex-degree vector V
F
(4)
i)1
0 1 D) 2 1 2
variables
]
RS ) [4.25, 4, 4.25, 4, 3] (f) The values of OEI, VDI, and RDI
OEI ) 1 × 12 - 0.25 × 8 ) 10.000000 VDI ) (7.75 × 8.25 × 7.75 × 8 × 5.75)1/5 ) 7.439830
-0.25 -0.25 -0.25 1 0
]
RDI ) (4.25 × 4 × 4.25 × 4 × 3)1/5 ) 3.869045 All the calculations can be performed quickly by computer. Likewise, the values of OEI, VDI, and RDI for 160 paraffins and 183 cycloalkanes were calculated and listed in Table 3. Obviously, the RDI index is zero for paraffins because of no cyclic atoms in their molecules. 3. RESULT AND DISCUSSION
In this section, we take 160 paraffins and 183 cycloalkanes (see Figure 3) with known boiling points (Bp) as the data
TOPOLOGICAL INDICES
J. Chem. Inf. Comput. Sci., Vol. 41, No. 4, 2001 869
Table 3. Variables N2/3, VDI, OEI, and RDI and Boiling Points (Bp) of 343 Paraffins and Cycloalkanes no.
compounda
N2/3
VDI
OEI
RDI
Bpexp
Bpcalc
∆
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73
1 2 3 4 2m3 5 2m4 22mm3 6 2m5 3m5 22mm4 23mm4 7 2m6 3m6 22mm5 23mm5 24mm5 33mm5 3e5 223mmm4 8 2m7 3m7 4m7 22mm6 23mm6 24mm6 25mm6 33mm6 34mm6 3e6 223mmm5 224mmm5 233mmm5 234mmm5 2m3e5 3m3e5 2233mmmm4 9 2m8 3m8 4m8 22mm7 23mm7 24mm7 25mm7 26mm7 33mm7 34mm7 35mm7 44mm7 3e7 4e7 223mmm6 224mmm6 225mmm6 233mmm6 234mmm6 235mmm6 244mmm6 334mmm6 2m3e6 2m4e6 3m3e6 3m4e6 2233mmmm5 2234mmmm5 2244mmmm5 2334mmmm5 22mm3e5 23mm3e5
1.0000 1.5874 2.0801 2.5198 2.5198 2.9240 2.9240 2.9240 3.3019 3.3019 3.3019 3.3019 3.3019 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267
0.0000 1.0000 2.1634 2.9131 3.3677 3.4010 3.8514 4.5895 3.7591 4.1451 4.1925 4.8758 4.6112 4.0367 4.3801 4.4146 5.0050 4.8234 4.7465 5.0972 4.4611 5.4937 4.2597 4.1290 4.6001 4.5940 5.1237 4.9583 4.9306 4.9008 5.1973 4.9992 4.6388 5.5893 5.4861 5.6310 5.3465 5.0075 5.2877 6.2627 4.4434 4.7280 4.7549 4.7479 5.2272 5.0775 5.0474 5.0562 5.0290 5.2906 5.1101 5.0849 5.2851 4.7903 4.7880 5.6355 5.5843 5.5546 5.6716 5.4611 5.3953 5.6199 5.7164 5.1211 5.0925 5.3702 5.1594 6.3158 6.0175 6.1265 6.0825 5.6903 5.7641
0.0000 2.0000 3.5000 5.2222 4.5000 6.8194 6.4444 5.0000 8.4967 7.9167 8.2639 7.1667 7.8889 10.1183 9.6739 9.8161 8.5139 9.5833 8.8889 9.2083 9.9583 8.8333 11.7808 11.2400 11.5178 11.3128 10.3511 11.2156 10.8683 10.9311 10.6356 11.3578 11.5906 10.4028 9.3611 10.7500 10.7778 11.1528 11.1250 10.0000 13.4120 12.9433 13.1247 13.0553 11.8617 12.8617 12.3094 12.7194 12.3061 12.4172 12.9344 12.7922 12.0072 13.2367 13.1672 12.1150 11.4206 11.6883 12.2572 12.6322 12.3478 11.5628 12.3994 12.8650 12.7228 12.6322 13.0072 11.7917 11.4722 9.7083 12.1667 11.8472 12.5417
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
-161.50 -88.60 -42.10 -0.50 -11.70 36.00 27.80 9.50 68.70 60.30 63.30 49.70 58.00 98.50 90.00 92.00 79.20 89.80 80.50 86.10 93.50 80.90 125.70 117.60 118.00 117.70 106.80 115.60 109.40 109.10 112.00 117.70 118.50 109.80 99.20 114.80 113.50 115.60 118.20 106.50 150.80 142.80 144.00 142.40 132.70 140.50 133.50 136.00 134.00 137.30 140.60 136.00 135.20 143.00 142.10 133.60 126.50 124.00 137.70 139.00 131.30 130.70 140.50 138.00 133.80 140.60 140.40 140.20 133.00 122.30 141.50 133.80 142.00
-140.31 -82.84 -43.66 -5.02 -14.47 30.95 23.66 7.77 64.46 57.54 58.40 45.64 51.86 95.07 89.56 89.72 77.98 84.24 82.39 79.67 89.76 73.65 123.40 122.81 118.90 118.22 109.08 114.03 113.05 113.59 109.35 114.12 118.76 104.27 101.47 105.12 108.28 113.28 110.19 95.40 149.42 145.03 145.38 145.21 136.31 141.28 139.67 141.00 139.84 137.61 141.21 140.97 136.24 145.42 145.20 133.07 131.14 132.39 133.20 136.64 136.31 131.28 133.25 140.86 140.65 137.56 140.97 124.96 126.89 119.42 128.69 131.57 133.27
-21.19 -5.76 1.56 4.52 2.77 5.05 4.14 1.73 4.24 2.76 4.90 4.06 6.14 3.43 0.44 2.28 1.22 5.56 -1.89 6.43 3.74 7.25 2.30 -5.21 -0.90 -0.52 -2.28 1.57 -3.65 -4.49 2.65 3.58 -0.26 5.53 -2.27 9.68 5.22 2.32 8.01 11.10 1.38 -2.23 -1.38 -2.81 -3.61 -0.78 -6.17 -5.00 -5.84 -0.31 -0.61 -4.97 -1.04 -2.42 -3.10 0.53 -4.64 -8.39 4.50 2.36 -5.01 -0.58 7.25 -2.86 -6.85 3.04 -0.57 15.24 6.11 2.88 12.81 2.23 8.73
870 J. Chem. Inf. Comput. Sci., Vol. 41, No. 4, 2001
CAO
AND
YUAN
Table 3 (Continued) no.
compounda
N2/3
VDI
OEI
RDI
Bpexp
Bpcalc
∆
74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147
24mm3e5 33ee5 10 2m9 3m9 4m9 5m9 22mm8 23mm8 24mm8 25mm8 26mm8 27mm8 33mm8 34mm8 35mm8 36mm8 44mm8 45mm8 3e8 4e8 223mmm7 224mmm7 225mmm7 226mmm7 233mmm7 234mmm7 235mmm7 236mmm7 244mmm7 245mmm7 246mmm7 255mmm7 334mmm7 335mmm7 344mmm7 345mmm7 2m3e7 2m4e7 2m5e7 3m3e7 3m4e7 3m5e7 4m3e7 4m4e7 4p7 4ip7 2233mmmm6 2234mmmm6 2235mmmm6 2244mmmm6 2245mmmm6 2255mmmm6 2334mmmm6 2335mmmm6 2344mmmm6 2345mmmm6 3344mmmm6 22mm3e6 22mm4e6 23mm3e6 23mm4e6 24mm3e6 24mm4e6 25mm3e6 33mm4e6 34mm3e6 33ee6 34ee6 2m3ip6 22334mmmmm5 22344mmmmm5 223mmm3e5 224mmm3e5
4.3267 4.3267 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416
5.4753 5.4625 4.5978 4.8603 4.8854 4.8789 4.8775 5.3165 5.1804 5.1517 5.1545 5.1625 5.1368 5.3739 5.2094 5.1821 5.1886 5.3668 5.2058 4.9189 4.9170 5.6842 5.6339 5.6424 5.6159 5.7158 5.5243 5.5000 5.4715 5.6651 5.5003 5.4401 5.6739 5.7544 5.7045 5.7542 5.5628 5.2211 5.1935 5.1965 5.4457 5.2563 5.2258 5.2532 5.4458 4.9170 5.2210 6.2890 6.0652 5.9987 6.1958 5.9764 6.1318 6.1259 6.0316 6.1026 5.8603 6.3744 5.7351 5.6808 5.9204 5.5749 5.5804 5.7488 5.5131 5.8100 5.8430 5.5336 5.3034 5.5425 6.6909 6.6071 6.3864 6.0856
12.2222 12.9167 15.0680 14.5433 14.7968 14.6309 14.7665 13.6058 14.5094 14.9027 14.2977 14.2316 14.1466 13.9686 14.6214 14.2741 14.5486 13.8297 14.5519 14.8844 14.8541 13.7056 12.8061 13.4211 12.8722 13.9833 14.1533 14.2161 14.0078 12.8789 14.0111 13.2506 13.6989 14.0561 13.5667 13.8511 14.4311 14.4556 14.2439 14.3828 14.3583 14.6639 14.5911 14.5283 14.0839 14.6883 14.5217 13.3789 13.4067 13.3272 11.9900 12.9800 12.5256 13.8961 13.2644 13.5489 13.9867 13.6633 13.6394 13.3550 15.0178 14.3617 14.1567 13.6394 14.0772 13.9239 14.2711 14.5039 14.7367 14.0144 13.0833 12.0417 13.4583 12.7917
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
136.70 145.00 174.10 166.80 167.80 165.70 165.10 155.00 164.00 153.00 157.00 158.50 160.00 161.20 166.00 159.00 159.50 160.00 162.40 168.00 164.00 158.00 147.70 147.00 148.20 160.00 163.00 159.70 155.70 152.00 157.00 145.00 152.80 164.00 155.70 164.00 164.00 166.00 160.00 159.70 163.80 167.00 160.00 167.00 166.00 161.80 159.50 160.00 155.00 148.70 153.00 147.90 137.00 164.00 153.00 162.00 158.00 170.50 159.00 147.00 169.00 164.00 164.00 158.00 157.00 165.00 170.00 166.30 162.00 163.00 166.00 159.30 168.00 155.30
135.07 137.62 173.56 169.44 170.02 169.55 170.00 162.10 166.33 167.88 165.89 165.60 165.56 162.74 166.43 165.55 166.39 162.36 166.23 169.99 169.91 158.93 156.42 158.38 156.81 159.54 161.93 162.36 161.95 156.36 161.68 159.75 159.00 159.41 158.27 158.74 162.47 165.78 165.35 165.77 163.34 166.12 166.17 165.71 162.44 169.38 165.99 152.01 154.27 154.64 148.22 153.70 150.66 155.32 154.12 154.38 158.17 152.14 158.22 157.79 161.01 162.13 161.40 158.09 161.78 158.45 159.28 162.99 165.92 161.29 147.08 144.35 151.33 152.01
1.63 7.38 0.54 -2.64 -2.22 -3.85 -4.90 -7.10 -2.33 -14.88 -8.89 -7.10 -5.56 -1.54 -0.43 -6.55 -6.89 -2.36 -3.83 -1.99 -5.91 -0.93 -8.72 -11.38 -8.61 0.46 1.07 -2.66 -6.25 -4.36 -4.68 -14.75 -6.20 4.59 -2.57 5.26 1.53 0.22 -5.35 -6.07 0.46 0.88 -6.17 1.29 3.56 -7.58 -6.49 7.99 0.73 -5.94 4.78 -5.80 -13.66 8.68 -1.12 7.62 -0.17 18.36 0.78 -10.79 7.99 1.87 2.60 -0.09 -4.78 6.55 10.72 3.31 -3.92 1.70 18.92 14.95 16.67 3.29
TOPOLOGICAL INDICES
J. Chem. Inf. Comput. Sci., Vol. 41, No. 4, 2001 871
Table 3 (Continued) no.
compounda
N2/3
VDI
OEI
RDI
Bpexp
Bpcalc
∆
148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221
234mmm3e5 2m33ee5 24mm3ip5 11 12 13 14 15 16 17 18 19 20 c3 c4 1mc3 bc110b c5 11mc3 12mc3 1ec3 1mc4 s22p mbc110b bc210p bc111p c6 123mc3 1e1mc3 1e2mc3 1pc3 1ipc3 112mc3 12mc4 13mc4 11mc4 1ec4 1mc5 bcpr 13mbcb s23hx bc220hx bc310hx mbc210p bc211hx c7 1sbc3 1122mc3 1123mc3 1tbc3 11ec3 12ec3 1e23mc3 1m2pc3 1m1ipc3 1bc3 1e3mc4 1e2mc4 1ipc4 1pc4 13mc5 1ec5 12mc5 11mc5 1mc6 dcprm 11ms22p 122mbcb s33h s24h 14mbc210p 13mbc111p mbc310hx 6mbc310hx
4.6416 4.6416 4.6416 4.9461 5.2415 5.5288 5.8088 6.0822 6.3496 6.6115 6.8683 7.1204 7.3681 2.0801 2.5198 2.5198 2.5198 2.9240 2.9240 2.9240 2.9240 2.9240 2.9240 2.9240 2.9240 2.9240 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.3019 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593
6.1787 5.8975 5.4747 4.7296 4.8436 4.9432 5.0312 5.1095 5.1796 5.2429 5.3002 5.3525 5.4004 4.0000 4.5000 4.8455 6.7454 5.0000 5.8730 5.4547 5.0057 5.1808 7.1895 7.4398 6.8770 6.8989 5.2222 5.9512 5.9087 5.5058 5.1137 5.6258 6.3022 5.7394 5.6499 6.0657 5.2935 5.5724 6.7458 7.9830 7.1692 6.9460 7.0281 7.5259 7.1598 5.4444 5.6938 7.0045 6.6900 6.4284 5.9675 5.5620 5.9457 5.5395 6.4073 5.2234 5.6901 5.7768 5.8221 5.3630 6.0090 5.6422 6.0699 6.3538 5.7200 6.6367 7.9764 8.1656 9.0714 7.3218 8.0749 7.9659 9.0489 7.2602
13.8333 14.2083 13.1667 16.7039 18.3563 19.9949 21.6453 23.2855 24.9346 26.5759 28.2241 29.8661 31.5136 6.0000 7.0000 7.0000 9.5000 7.5000 7.5000 8.2222 8.9444 8.2222 10.0000 10.0000 10.0000 10.0000 9.6667 9.6667 9.6667 10.0417 10.4167 10.3889 8.9444 9.6667 9.3194 8.9444 10.0417 8.9444 12.8889 10.7222 11.4444 11.4444 10.7222 10.7222 10.0000 12.0556 12.0833 9.8889 10.6111 11.3333 11.7083 11.9411 11.3611 11.5939 11.3333 12.1739 11.2189 11.3611 11.3611 11.5939 10.2639 10.6389 10.6111 9.8889 10.9861 13.8889 12.3889 12.3889 12.7639 12.3889 11.6667 11.3194 11.6667 12.0417
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2.0000 2.2500 1.8612 4.1231 2.5000 1.7826 1.7826 1.4147 2.1046 3.5947 3.8690 4.0889 4.0696 2.6111 1.7321 1.4286 1.4286 1.0497 1.1783 1.7321 2.0129 2.0129 2.0129 1.6749 2.3494 2.7760 3.7084 3.5846 4.0633 4.0852 3.9069 4.1457 2.7222 0.9365 1.6968 1.6969 1.0340 1.2196 1.2196 1.4386 1.1046 1.2196 0.7892 1.6657 1.6657 1.4228 1.2946 2.2475 1.9319 2.2475 2.2475 2.4730 2.3998 3.0282 3.2541 3.5947 3.6609 3.7818 3.7202 3.9490 3.7597
169.40 174.00 157.00 195.90 216.30 235.40 253.70 270.60 287.00 301.80 316.10 329.70 343.00 -32.80 12.60 0.70 8.00 49.30 20.60 32.60 35.90 36.30 39.00 33.50 46.00 36.00 80.70 63.00 57.00 63.00 69.00 58.30 52.60 62.00 59.00 53.60 70.70 71.80 76.00 55.00 69.50 83.00 81.00 60.50 71.00 118.40 90.30 76.00 78.00 80.50 88.60 90.00 91.00 93.00 81.50 98.00 89.50 94.00 92.70 100.70 91.30 103.50 95.60 87.90 101.00 102.00 78.00 84.00 96.50 98.50 74.00 71.50 92.00 103.00
154.60 158.55 159.14 195.86 216.62 235.90 253.90 270.68 286.40 301.10 314.90 327.85 340.03 -34.87 5.83 -2.79 7.31 38.75 20.68 29.12 34.18 35.72 33.18 32.79 41.89 41.44 76.32 59.69 57.32 63.61 66.19 61.27 52.53 64.81 64.45 58.02 68.39 66.97 73.16 58.61 69.93 76.89 73.27 65.85 69.42 110.61 92.54 76.04 82.46 82.28 90.56 95.94 91.41 93.85 84.18 96.77 95.68 95.27 92.64 97.46 93.63 96.33 94.29 88.36 101.51 101.85 86.86 86.76 81.08 99.68 89.68 89.00 80.22 99.89
14.80 15.45 -2.14 0.04 -0.32 -0.50 -0.20 -0.08 0.60 0.70 1.20 1.85 2.97 2.07 6.77 3.49 0.68 10.55 -0.08 3.48 1.72 0.58 5.82 0.71 4.11 -5.44 4.38 3.31 -0.32 -0.61 2.81 -2.97 0.07 -2.81 -5.45 -4.42 2.31 4.83 2.84 -3.61 -0.43 6.11 7.73 -5.35 1.58 7.79 -2.24 -0.04 -4.46 -1.78 -1.96 -5.94 -0.41 -0.85 -2.68 1.23 -6.18 -1.27 0.06 3.24 -2.33 7.17 1.31 -0.46 -0.51 0.15 -8.86 -2.76 15.42 -1.18 -15.68 -17.50 11.78 3.11
872 J. Chem. Inf. Comput. Sci., Vol. 41, No. 4, 2001
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Table 3 (Continued) no.
compounda
N2/3
VDI
OEI
RDI
Bpexp
Bpcalc
∆
222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295
2mbc310hx bc320h mbc211hx tc221026h bc221h bc410h bc311h tc410024h tc410013h tc410027h tec410h tc311024h tec320h c8 11223mc3 11m2pc3 ib2mc3 112m2ec3 11m2ipc3 1nepec3 p2ec3 b2mc3 1spec3 5msbc3 1pec3 1m12ec3 1133mc4 1234mc4 12ec4 1sbc4 p3mc4 123mc5 124mc5 1e1mc5 1e2mc5 1e3mc5 113mc5 112mc5 1pc5 1ipc5 14mc6 13mc6 12mc6 11mc6 1ec6 1mc7 bcprm bcb s34o 1223mbcb 2244mbcb 33mbc310hx 2mbc320h bc330o s25o 1mbc410h 7mbc410h bc420o 1mbc221h 2mbc221h 7mbc221h 1mtc2210h ds2121o ds2022o tc3210o 3mtc2210h bc510o tc510035o tc510024o tc3300o tec330o 14mbc211hx c9 1shxc3
3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 3.6593 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.3267 4.3267
7.3264 7.1288 7.7347 8.9764 7.3287 7.0013 7.1163 8.6910 8.9775 8.4629 10.2560 8.8149 10.4756 5.5694 7.3293 6.1775 5.9744 6.9210 6.6228 6.3372 5.5961 5.5918 5.7208 6.1506 5.3212 6.2566 6.9429 6.5416 5.8236 5.8745 5.7092 6.4688 6.4124 6.3768 6.0880 6.0265 6.6696 6.7733 5.6741 6.0981 6.1117 6.1234 6.1701 6.4157 5.7813 5.8868 6.5825 6.8454 7.3485 8.6346 8.1535 7.8133 7.4084 7.2920 7.2621 7.5600 7.2421 7.0977 7.8877 7.5717 7.6195 9.3442 8.7535 8.9710 8.8162 9.0429 7.0172 8.5201 8.5428 8.8808 10.0389 8.1835 5.6944 5.7586
12.3889 12.3889 10.9444 12.0000 10.9444 13.1111 12.3889 14.1667 13.4444 14.8889 15.9444 13.4444 14.5000 13.2778 11.7778 12.2711 12.6461 11.8056 11.8333 11.8611 13.4378 13.2956 13.7156 13.2778 13.7400 12.7606 10.3889 12.5278 13.1356 13.1356 12.7156 12.1528 11.8056 11.8056 12.1806 12.0383 11.0833 11.7778 12.2711 11.8333 12.3856 12.1806 12.5278 11.8056 12.7606 13.2500 15.9311 14.9133 13.5833 13.5556 12.8889 12.8611 13.9306 13.2083 14.3056 13.9306 14.3056 14.3056 12.1111 12.4861 12.8333 13.1667 15.3889 16.7778 14.6111 13.8889 15.3750 16.4306 16.7778 13.8889 14.4167 11.7639 14.3750 15.3617
3.7381 4.0974 3.9740 5.5476 4.1947 3.9912 4.0512 5.3986 5.6588 5.2045 6.7084 5.4827 6.8399 2.7847 1.6709 1.1477 0.9109 1.4461 1.2516 0.7229 0.9933 0.8552 0.7401 0.7883 0.6061 1.2516 1.9039 1.9039 1.4451 1.1592 1.3305 2.1739 2.1739 1.9044 1.9044 1.9044 2.1739 2.1739 1.5407 1.6682 2.3742 2.3742 2.3742 2.3742 2.0874 2.5934 2.1212 2.9490 3.6742 3.2212 2.9503 3.4201 3.8030 4.1365 3.6310 3.8952 3.7349 4.0175 4.0543 3.8820 3.9209 5.2728 4.3767 4.4855 5.3879 5.0622 3.9425 5.1907 5.2149 5.4327 6.6195 3.8499 2.8472 0.5895
100.00 110.50 81.50 106.00 105.50 116.00 110.00 105.00 107.50 110.00 104.00 107.00 108.50 149.00 100.50 105.90 110.00 104.50 94.40 106.00 108.00 124.00 117.70 115.50 128.00 108.90 86.00 114.50 119.00 123.00 117.40 117.00 115.00 121.50 124.70 121.00 104.90 114.00 131.00 126.40 121.80 122.30 126.60 119.50 131.80 134.00 129.00 136.00 128.00 105.00 104.00 115.00 130.50 137.00 125.00 125.00 138.00 133.00 117.00 125.00 128.00 111.00 103.00 115.00 136.00 120.50 141.00 142.00 149.00 125.00 137.50 91.00 175.00 143.00
100.30 105.54 92.38 96.55 98.78 108.69 105.28 106.52 102.97 109.99 107.54 103.21 100.89 137.97 105.01 114.68 116.20 107.58 109.21 107.86 123.70 122.09 121.31 115.62 124.44 116.51 105.99 118.13 123.99 121.10 122.72 119.78 119.11 117.23 121.61 121.74 113.77 115.22 123.23 118.28 125.98 125.13 125.89 120.75 128.36 133.13 131.68 132.13 128.19 110.87 111.23 118.68 129.86 131.21 131.28 129.06 132.33 136.07 120.47 123.68 124.75 119.22 125.94 129.54 130.79 123.22 139.99 138.57 139.74 127.94 127.64 114.38 162.43 147.76
-0.30 4.96 -10.88 9.45 6.72 7.31 4.72 -1.52 4.53 0.01 -3.54 3.79 7.61 11.03 -4.51 -8.78 -6.20 -3.08 -14.81 -1.86 -15.70 1.91 -3.61 -0.12 3.56 -7.61 -19.99 -3.63 -4.99 1.90 -5.32 -2.78 -4.11 4.27 3.09 -0.74 -8.87 -1.22 7.77 8.12 -4.18 -2.83 0.71 -1.25 3.44 0.87 -2.68 3.87 -0.19 -5.87 -7.23 -3.68 0.64 5.79 -6.28 -4.06 5.67 -3.07 -3.47 1.32 3.25 -8.22 -22.94 -14.54 5.21 -2.72 1.01 3.43 9.26 -2.94 9.86 -23.38 12.57 -4.76
TOPOLOGICAL INDICES
J. Chem. Inf. Comput. Sci., Vol. 41, No. 4, 2001 873
Table 3 (Continued)
a
no.
compounda
N2/3
VDI
OEI
RDI
Bpexp
Bpcalc
∆
296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343
1123mc5 1122mc5 1133mc5 1m1pc5 12m1ec5 13ec5 12ec5 11ec5 1ibc5 1m3ipc5 1tbc5 3pec4 1bc5 1m2pc5 1sbc5 11m2ec5 11m3ec5 113mc6 124mc6 135mc6 1e2mc6 1e3mc6 1pc6 1ipc6 112mc6 114mc6 1m1ec6 1m4ec6 1ec7 12mc7 2ebc221h 4ms25o bc430n dc4m 15ms33h 77mbc221h c10 13ec6 1m2ipc6 1m3ipc6 1m4ipc6 1bc6 1sbc6 1ibc6 1tbc6 1pc7 26mbc321o 37mbc330o
4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.3267 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416 4.6416
7.0783 7.4014 7.2260 6.3273 6.7674 6.0513 6.1199 6.4247 6.0651 6.4119 6.7312 5.9404 5.7164 6.0757 6.1287 6.6684 6.6163 6.7302 6.4935 6.4600 6.1929 6.1449 5.8046 6.1888 6.8104 6.7084 6.4472 6.1284 5.9394 6.2983 7.4064 7.5723 7.2729 6.7582 7.7325 8.1707 5.7744 6.1727 6.5461 6.4969 6.4772 5.8361 6.2176 6.1551 6.7632 5.9537 7.7939 7.6516
13.1944 13.1667 11.7778 13.3128 13.5694 13.7572 13.8300 13.5972 13.4033 13.3128 12.5278 14.7850 13.9172 13.8928 13.6878 13.2847 12.9378 12.8750 13.8022 13.2500 14.1772 14.0350 14.3372 14.0350 13.5694 13.2850 13.8022 14.1044 15.1044 14.6667 14.1356 16.0694 15.0000 16.3478 15.7222 14.2222 16.3722 15.8339 15.3267 15.3894 15.3233 16.0241 15.8339 15.4139 14.8094 16.6256 16.4689 12.4308
2.1184 2.1184 2.1184 1.5599 1.8832 1.6741 1.6741 1.6741 1.2921 1.6741 1.4883 0.9884 1.2267 1.5599 1.3867 1.8832 1.8832 2.3001 2.3001 2.3001 2.0514 2.0514 1.7084 1.8296 2.3001 2.3001 2.0514 2.0514 2.2340 2.4975 3.2890 2.8388 4.0756 2.6271 3.2053 3.7204 2.8872 1.8250 1.8250 1.8250 1.8250 1.3908 1.5479 1.4553 1.6464 1.8672 3.6762 3.6153
134.00 135.00 118.20 145.00 142.50 148.20 150.50 151.00 148.00 141.00 145.00 148.70 156.60 149.50 154.30 138.00 133.00 136.60 144.80 139.50 154.30 150.00 156.70 154.80 145.10 136.00 152.00 150.80 163.70 157.00 152.00 149.00 163.00 161.80 132.20 148.20 202.00 172.00 171.00 167.00 170.00 180.90 179.30 171.30 171.50 182.80 164.50 166.00
139.32 135.97 132.88 142.77 141.84 147.91 147.49 143.69 143.57 142.84 135.44 147.13 148.24 147.24 144.66 141.85 141.17 143.12 148.62 147.07 150.89 150.88 152.55 148.72 144.71 144.75 147.13 151.27 157.85 154.96 148.53 149.95 158.79 157.13 150.07 144.69 187.84 174.89 169.81 170.48 170.44 175.41 172.44 170.99 164.77 179.71 175.58 163.42
-5.32 -0.97 -14.68 2.23 0.66 0.29 3.01 7.31 4.43 -1.84 9.56 1.57 8.36 2.26 9.64 -3.85 -8.17 -6.52 -3.82 -7.57 3.41 -0.88 4.15 6.08 0.39 -8.75 4.87 -0.47 5.85 2.04 3.47 -0.95 4.21 4.67 -17.87 3.51 14.16 -2.89 1.19 -3.48 -0.44 5.49 6.86 0.31 6.73 3.09 -11.08 2.58
The molecular structures of compounds are showed in Figure 3. Compounds of no. 151-160 come from ref 9; the others come from ref 10.
set to test the QSPR applicability of the three indices, OEI, VDI, and RDI, in combination with another variable N2/3 (N is the number of carbon atoms. N2/3 is a useful variable correlating with the boiling points.8 So N2/3 is used here.). The following correlations have been carried out
Bp ) a0 + a1N2/3 + a2VDI + a3OEI + a4RDI (6a) ln(Bp0 - Bp) ) b0 + b1N2/3 + b2VDI + b3OEI + b4RDI (7a) where a0 and b0 are the intercept terms; a1, a2, a3, a4, b1, b2, b3, and b4 are the regression coefficients; and Bp0 ) 577, which is a parameter obtained by an optimization technique with the purpose of getting the lowest standard deviation for the examined compounds.
Figure 2. Plot of Bpcalc vs Bpexp for 343 compounds.
874 J. Chem. Inf. Comput. Sci., Vol. 41, No. 4, 2001
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TOPOLOGICAL INDICES
J. Chem. Inf. Comput. Sci., Vol. 41, No. 4, 2001 875
Figure 3. The molecular structures of 160 paraffins and 183 cycloalkanes in Table 3.
876 J. Chem. Inf. Comput. Sci., Vol. 41, No. 4, 2001
According to the regression models 6a and 7a, we obtained the corresponding eqs 6b and 7b, respectively:
Bp ) -182.821198 + 77.491737N2/3- 4.904214VDI + 0.731171OEI + 6.742943RDI (6b) F ) 2650.74, r ) 0.9844, rms ) 9.12 °C, n ) 343 ln(577 - Bp) ) 6.729609 - 0.154107N2/3 + 2.285632 × 10-2VDI - 7.921410 × 10-3OEI 1.821962 × 10-2RDI (7b) F ) 6455.09, r ) 0.9935, rms ) 6.44 °C, n ) 343 where rms is the root-mean-square error, rms ) M (Bpexp-Bpcalc)2/(M-1), in which M is the number of x∑i)1
examined compounds. In comparison, eq 7b gets much better correlation result than eq 6b, which shows that the linear model does not reflect the exact relationship between Bp and the four variables. Many examinations have revealed that the plot of Bp vs N2/3 observes a logarithm function. Therefore, we prefer the logarithm transformation mode of Bp as the dependent variable instead of Bp.6 The high correlation coefficient and low rms error show that eq 7b is a good model for estimating or predicting the boiling points of alkanes. Furthermore it employed only four variables for such a large and combined sample and the computation of these variables is simple. So we may say that these four variables have expressed most of the molecular structure information and can be used for QSPR study. Before reaching the above conclusion, this paper contains the four variables which have been studied in the following several aspects: (1) In general, the variables employed in one regression equation should be independent of each other, that is to say, the molecular structure information can be expressed by fewer variables instead of more variables with high linear correlation among them. So the interrelation of variables is an important mark to evaluate the rationality of the defined indices. In this paper, we studied the interrelations of these four variables (listed in Table 1) and found that N2/3 is closely related to OEI. But for the isomers with the main carbon chain of the same length, the values of N2/3 are equal, while the values of OEI vary in a wavy line with the variation of the substituted situation (for instance, n-methylnonane, when n ) 2, 3, 4, 5, their N are all 10 and their OEI are 14.5433, 14.7968, 14.6309, 14.7665, respectively), which indicates that OEI can reflect more details about the molecular structure features than N2/3. So OEI is an important molecular descriptor and cannot be replaced by N2/3 in the Bp-structure study. (2) To evaluate the importance of each variable in the QSPR study and the contribution of the number of variables to the result, we regressed one to four variables (N2/3, VDI, OEI, RDI) against the boiling points of 343 hydrocarbons as to model 7a, respectively. The obtained results are listed in Table 2. As shown in Table 2, the single-variable regression of N2/3 has a high correlation coefficient (r ) 0.9761), which proves that Bp is closely related to N2/3. The poor correlation results obtained by regressions with VDI, RDI, and OEI without N2/3 also indicate that N2/3 is very important in Bp-structure
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Table 4. Comparison of the Results from the Recent Literatures with Ours on the Bp-Structure Studies for Paraffins and Cycloalkanes compounds paraffins cycloalkanes
d
NCa 150 80
variables
NVb
r
rms (°C)
x′ λ11,‚‚‚, λ44d N2/3, VDI, OEIe µ0,‚‚‚ µ7f N2/3, VDI, OEI, RDIe
1 10 3 6 4
0.9945 0.9954 0.9952 0.9937 0.9955
5.15 4.82 4.21 4.80 4.78
c
a Number of compounds. b Number of variables. c From ref 11. From ref 6. e From this work. f From ref 7.
studies. But only one variable, N2/3, is not enough to deliver most of the molecular structure information. It should be in combination with other variables VDI, RDI, and OEI to express more details about the structure features. The fourvariable regression yields the best result (see eq 7b), by which the calculated Bps(Bpcalc) and the differences (∆ ) Bpexp Bpcalc) between Bpexp and Bpcalc are listed in Table 3. The linear relationship between the experimental and calculated boiling points for 343 compounds is given in Figure 2. (3) In comparison, in this paper are collected several previous literatures on the study of Bp-structure relationships for alkanes and made regression analysis against various samples with the newly defined indices. The results of each case were listed in Table 4. 4. CONCLUSION
People have been going into QSPR/QSAR studies with more or less different variables, but so far, there is not a wide-admitted criterion about the number of variables employed in a QSPR model. For different samples and properties, it is difficult to define a specific number of variables. However a common rule should be complied with that the variables employed not only can render most of the molecular structure information but also not seem to be burdensome. In the Bp-structure regression for a large combined sample of acyclic, monocyclic, and polycyclic alkanes with N2/3, OEI, VDI, and RDI, the correlation coefficient is good and the root-mean-square error is within the reasonable limits, which proves that the newly defined indices based on the essential elements of molecular graph are useful in QSPR study and worthy of further investigation. ACKNOWLEDGMENT
This project was supported by Hunan Province Education Commission (HPEC). REFERENCES AND NOTES (1) Randic, M. On Characterization of Molecular Branching. J. Am. Chem. Soc. 1975, 97, 6609-6615. (2) Balaban, A. T. Highly Discriminating Distance-Based Topological Index. Chem. Phys. Lett. 1982, 89, 399-404. (3) Schultz, H. P. Topological Organic Chemistry. 1. Graph Theory and Topological Indices of Alkanes. J. Chem. Inf. Comput. Sci. 1989, 29, 227-228. (4) Wiener, H. Structural Determination of Paraffin Boiling Points. J. Am. Chem. Soc. 1947, 69, 17-20. (5) Mihalic, Z.; Trinajstic, N. A Graph-Theoretical Approach to StructureProperty Relationships. J. Chem. Educ. 1992, 69, 701-712.
TOPOLOGICAL INDICES (6) Liu, S.; Cao, C.; Li, Z. Approach to Estimation and Prediction for Normal Boiling Point (NBP) of Alkanes Based on a Novel Molecular Distance-Edge (MDE) Vector, λ. J. Chem. Inf. Comput. Sci. 1998, 38, 387-394. (7) Estrada, E. Spectral Moments of the Edge Adjacency Matrix in Molecular Graphs. 3. Molecules Containing Cycles. J. Chem. Inf. Comput. Sci. 1998, 38, 23-27. (8) Cao, C.; Liu, S.; Li, Z. On Molecular Polarizability: 2. Relationship to the Boiling Point of Alkanes and Alcohols. J. Chem. Inf. Comput. Sci. 1999, 39, 1105-1111.
J. Chem. Inf. Comput. Sci., Vol. 41, No. 4, 2001 877 (9) CRC Handbook of Chemistry and Physics; 58th ed.; Weast, R. C., Ed.; CRC Press: 1977-1978; c280-c532. (10) Ru¨cker, G.; Ru¨cker, C. On Topological Indices, Boiling Points, and Cycloalkanes. J. Chem. Inf. Comput. Sci. 1999, 39, 788-802. (11) Mihalic, Z. Comparative Study of Molecular Descriptors Derived from the Distance Matrix. J. Chem. Inf. Comput. Sci. 1992, 32, 28-37.
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