J. Chem. InJ Comput. Sci. 1994,34, 1140-1 145
1140
Extended Adjacency Matrix Indices and Their Applications
Yi-Qiu Yang, Lu XU,' and Chang-Yu H u
Applied Spectroscopy Laboratory, Changchun Institute of Applied Chemistry, Academia Sinica,
Changchun 130022, Jilin, People's Republic of China
Received March 30, 1994'
In this paper, new topological indices, EAC and EAmax, are introduced. They are based on the extended
adjacency matrices of molecules, in which the influences of factors of heteroatoms and multiple bonds were
considered. The results show that EAC and EAmax possess high discriminating power and correlate well with
a number of physicochemical properties and biological activities of organic compounds.
INTRODUCTION
The structure of a molecule-geometric
and electronic-must contain the features responsible for its physical
and chemical properties. The simplest way to represent a
molecule's structure is to assign to the structure a number or
a set of numbers, termed indices; then, the indices are applied
to the study of the correlations with properties ranging from
physical to biological. In recent years, one type of these
methods seems to hold good promise for the quantitative
structure-property relationship (QSPR) and the quantitative
structure-activity relationship (QSAR) studies.
The topological indices are actually graph invariants. They
are obtained from the chemical graph (hydrogen-suppressed
graph). Uniqueness and correlation are two of the most
important requirements for the topological indices. The task
of defining an index that could have different, unique, but
structurally significant, values for different structures seems
to be very difficult. Therefore, more than 100 topological
indices are in existence, such as the Wiener index, W,'the
~ Balaban index,
Hosoya index, Z,2 the RandiC index x , the
5,4and the extended RandiC index by Kier and Hall.s In this
paper, the new indices, EA'S proposed by us, show lower
degeneracy and good performance in correlation.
METHODS
Recently, RandiC discussed the strategies for searching
optimal molecular de~criptors.6~~
As we know already, most
indices to date have been based on two particular matrices:
the distance and adjacency matrices. In this paper, the
adjacency matrix and its extended form were used to deduce
the new topological indices. The entries in the adjacency
matrix are symbolized as av and are equal to either one or
zero, depending respectively on whether or not the vertices
are connected:
1 for adjacent vertices
aij =
0 otherwise
For example the hydrogen-suppressed graph of 2-methylbutane is
Table 1. Electronegativities of Common Atoms
H
2.1
C
N
3.0
0
F
4.0
P
S
2.5
CI
Br
2.8
I
2.5
3.5
2.1
3.0
2.5
Table 2. Values of ID, EAC, and EAmax of 20 Structures
no.
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
ID
18.0379058
24.315 9362
15.0076026
19.2052889
19.2127979
19.3869551
26.868 1152
7.777 7778
17.991 7696
20.263 3
22.4172175
22.5248849
22.457 3998
17.6762689
22.1002019
21.417 0255
31.2107297
21.3924287
21.178 7608
19.4773775
EA1
14.798 27
19.65951
10.461 1 1
13.375 89
13.11984
14.105 44
17.44643
6.000000
12.00000
13.29879
14.285 75
14.71333
15.517 54
12.292 53
15.30758
14.49962
21.656 12
15.121 58
16.19804
12.13982
EAmax
3.337 117
3.337 117
2.936060
2.653 472
2.672 798
2.829541
2.829 542
3.000 000
3.000000
2.951932
2.914090
2.906 148
3.000000
3.OOO 000
2.914090
2.653 614
2.777381
2.620 189
3.049510
3.005782
Table 3. EA Indices for 18 Graphs
no.
EAC
1 1.25963
11.49126
1 2.21042
11.551 54
11.55315
11.57474
11.42910
1 1.64761
10.68483
EAmax
2.878785
2.864037
2.822746
2.865975
2.824196
2.822746
2.822746
2.884355
2.884355
no.
10
11
12
13
14
15
16
17
18
EAz
11.65685
1 1.73486
1 1.73486
11.32856
11.32856
12.47214
12.47214
EAmax
3.000000
2.885716
2.885716
2.894902
2.894902
3 .000000
3.oooo00
12.00000
3 .OOOOOO
12.00000
3.000000
and its adjacency matrix takes the form
n
Abstract published in Aduance ACS Absrracrs, July 15, 1994.
0095-233819411634-1 140S04.50,IO
I
,
We also introduce the degree vector V, V = (vi), where ui
0 1994 American Chemical Society
J. Chem. In!
EXTENDED
ADJACENCY
MATRIXINDICES
Comput. Sci., Vol. 34, No. 5, I994 1141
Table 4. Towlogical Indices and Boiling Points for 149 Alkanes
~
~~
compounds
ethane
propane
butane
2-methylpropane
pentane
2-methylbutane
2,2-dimethylpropane
hexane
2-methylpentane
3-methylpentane
2,3-dimethylbutane
2,2-dimethylbutane
heptane
2-methylhexane
3-methylhexane
3-ethylpentane
2,4-dimethylpentane
2,3-dimethylpentane
2,2-dimethylpentane
3,3-dimethylpentane
2,2,3-trimethylbutane
octane
2methylheptane
3-methylheptane
4-methylheptane
3-ethylhexane
2,5-dimethylhexane
2,4-dimethylhexane
2,3-dimethylhexane
3-ethyl-2-methylpentane
3,4-dimethylhexane
2,2-dimethylhexane
3,3-dimethylhexane
3-ethyl-3-methylpentane
2,3,4-trimethylpentane
2,2,4-trimethylpentane
2,2,3-trimethylpentane
2,3,3-trimethylpentane
2,2,3,3-trimethylbutane
nonane
2-methyloctane
3-methyloctane
4-methyloctane
4-ethylheptane
3-ethylheptane
2,6-dimethylheptane
2,5-dimethylheptane
2,3-dimethylheptane
2,4-dimethylheptane
3-ethyl-2-methylhexane
4-ethyl-2-methylhexane
3,5-dimethylheptane
3,4-dimethylheptane
3-ethyl-4-methylhexane
2,2-dimethylheptane
3,3-dimethylheptane
4,4-dimethylheptane
4,4-diethylpentane
3-ethyl-3-methylhexane
2,3,5-trimethylhexane
2,4-dimethyl-3-ethylpentane
2,3,4-trimethylhexane
2,2,5-trimethylhexane
2,2,4-trimethylhexane
2,2,3-trimethylhexane
2,2-dimethyl-3-ethylpentane
2,4,4-trimethylhexane
2,3,3-trimethylhexane
3,3,4-trimethylhexane
2,3-dimethyl-3-ethylpentane
2,2,3,4-tetramethylpentane
2,3,3,4-tetramethylpentane
2,2,4,4-tetramethylpentane
2,2,3,3-tetramethylpentane
decane
2-methylnonane
3-methylnonane
W
Z
X
J
1
4
10
9
20
18
16
35
32
31
29
28
56
52
50
48
48
46
46
44
42
84
79
76
75
72
74
71
70
67
68
71
67
64
65
66
63
62
58
120
114
110
108
102
104
108
104
102
102
96
98
100
98
94
104
98
96
88
92
96
90
92
98
94
92
88
92
90
88
86
86
84
88
82
165
158
153
2
3
5
4
8
7
5
13
11
12
10
9
21
18
19
20
15
17
14
16
13
34
29
31
30
32
25
26
27
28
29
23
25
28
24
19
22
23
17
55
47
1.000 000
1.414214
1.914 214
1.732 05 1
2.414 214
2.270 056
2.000 000
2.914 214
2.770 056
2.808 061
2.642 735
2.560 660
3.414 214
3.270 056
3.308 061
3.346 066
3.125 898
3.180 739
3.060 660
3.121 320
2.943 376
3.914 214
3.770 056
3.808 061
3.808 061
3.846 066
3.625 898
3.663 903
3.680 739
3.718 744
3.718 744
3.560 660
3.621 320
3.681 981
3.553 418
3.416 502
3.481 381
3.504 036
3.250 000
4.414 214
4.270 056
4.308 061
4.308 061
4.346 066
4.346 066
4.125 898
4.163 903
4.180 739
4.163 903
4.218 744
4.201 908
4.201 908
4.218 744
4.256 749
4.060 660
4.121 321
4.121 321
4.242 640
4.181 981
4.036 582
4.091 423
4.091 423
3.916 502
3.954 507
3.981 381
4.019 385
3.977 163
4.004 036
4.042 041
4.064 696
3.854 059
3.886 752
3.707 107
3.810 660
4.914 214
4.770 056
4.808 061
1.ooo 000
50
49
51
52
40
43
44
41
45
44
45
46
48
37
41
39
48
44
37
39
41
32
33
35
36
34
36
39
40
31
33
24
30
89
76
81
1.632 993
1.974 745
2.323 790
2.190 610
2.539 539
3.023 716
2.339 092
2.627 215
2.754 185
2.993 498
3.168 490
2.447 473
2.678 258
2.831 820
2.992 303
2.953 223
3.144 208
3.154 490
3.360 435
3.541 197
2.530 061
2.7 15 843
2.872 066
2.919 613
3.074 373
2.927 820
3.098 828
3.180 819
3.354 877
3.292 478
3.111 766
3.373 382
3.583 212
3.464 227
3.388 924
3.623 28 1
3.708 324
4.020 391
2.595 083
2.746 691
2.876 623
2.954 823
3.175 341
3.092 246
2.914 659
3.060 821
3.155 280
3.151 251
3.410 085
3.307 394
3.223 047
3.324 760
3.499 480
3.072 990
3.330 074
3.431 051
3.824 684
3.617 390
3.376 601
3.677 616
3.575 834
3.280 71 1
3.467 262
3.588 734
3.792 908
3.576 752
3.702 086
3.802 396
3.919 211
3.877 605
4.013 737
3.746 418
4.144 726
2.647 605
2.773 189
2.886 163
EAE
2.000 000
3.535 534
5.385 165
5.773 502
6.274 918
7.532 390
8.500 001
7.875 304
8.320 846
9.089 133
9.637 889
10.173 18
8.892 971
9.992 3 15
9.929 288
9.509 251
10.336 42
11.16928
10.929 39
11.715 59
12.253 66
10.396 22
10.968 48
11.564 35
10.752 93
11.045 75
12.116 62
11.960 89
12.014 14
11.470 13
12.707 51
12.621 92
12.491 21
13.019 51
13.244 01
12.934 07
13.776 70
13.777 61
14.857 66
11.477 15
12.504 77
12.561 53
12.400 64
12.048 32
12.103 02
13.035 63
13.685 03
13.647 04
12.779 65
13.072 26
13.155 79
13.576 83
13.551 02
13.045 09
13.586 60
14.171 24
13.262 69
13.090 17
13.840 58
14.047 27
13.398 14
14.783 69
14.748 63
14.564 33
14.622 88
14.046 11
14.502 35
14.554 98
15.303 75
15.050 23
15.849 47
15.835 85
15.527 46
16.375 59
12.928 00
13.563 13
14.081 01
EAmax
1.ooo 000
1.767 767
1.846 291
2.886 751
1.887 459
2.657 551
4.250 000
1.912 646
2.644 6 17
2.441 140
2.909 472
3.909 893
1.929 549
2.641 847
2.428 892
2.254 625
2.811 188
2.786 288
3.902 909
3.549 463
3.905 577
1.941 614
2.641 261
2.425 542
2.416 806
2.246 364
2.722 244
2.724 521
2.780 567
2.693 850
2.631 516
3.902 374
3.540 540
3.172 473
2.919 351
3.914 430
3.877 758
3.568 858
4.214 415
1.950 614
2.641 138
2.424 473
2.413 346
2.238 559
2.243 288
2.679 947
2.664 994
2.779 527
2.721 230
2.689 473
2.680 721
2.580 399
2.623 622
2.512 713
3.902 333
3.539 675
3.531 518
2.795 085
3.161 112
2.848 745
2.828 666
2.842 428
3.903 268
3.910 517
3.877 049
3.855 443
3.561 424
3.560 822
3.527 114
3.239 333
3.882 490
3.585 167
4.083 121
4.054 641
1.957 555
2.641 112
2.424 202
bP ("C
-88.6
-42.1
-0.50
-1 1.73
36.07
27.85
9.50
68.74
60.27
63.28
57.99
49.74
98.42
90.05
91.85
93.48
80.50
89.78
79.20
86.03
80.88
125.67
117.65
118.93
117.71
118.54
109.10
109.43
115.61
115.65
117.72
106.84
111.97
118.25
113.46
99.23
109.84
114.76
106.47
150.8
142.8
143.5
142.4
141.2
143.0
135.2
136.0
140.5
133.5
138.0
133.8
136.0
140.1
140.4
132.7
137.3
135.2
146.2
140.6
131.3
136.73
139.0
124.0
126.5
131.7
133.83
126.5
137.7
140.5
141.6
133.0
141.5
122.7
140.3
174.2
167
167.8
1142 J. Chem. If. Comput. Sci., Vol. 34, No. 5, 1994
YANG ET AL.
Table 4 (Continued)
compounds
4-methylnonane
5-methylnonane
4-isopropylheptane
4-ethyloctane
3-ethyloctane
2,7-dimethyloctane
2,6-dimethyloctane
2,3-dimethyloctane
2,5-dimethyloctane
2,4-dimethyloctane
4-isopropylheptane
4-ethyl-2-methylheptane
3-ethyl-2-methylheptane
5-ethyl-2-methylheptane
3,6-dimethyloctane
3,4-dimethyloctane
3,5-dimethyloctane
4,5-dimethyloctane
4-ethyl-3-methylheptane
5-ethyl-3-methylheptane
3-ethyl-4-methylheptane
3,4-diethylhexane
2,2-dimethyloctane
3,3-dimethyloctane
4,4-dimethyloctane
3,3-diethylhexane
4-ethyl-4-methylheptane
3-ethyl-3-methylheptane
2,3,6-trimethylheptane
2,4,6-trimethylheptane
3-isopropyl-2-methylhexane
2,5-dimethyl-3-ethylhexane
2,3,5-trimethylheptane
2,4,5-trimethylheptane
2,3,4-trimethylheptane
2,4-dimethyl-3-ethylhexane
2,3-dimethyl-4-ethylhexane
3,4,5-trimethylheptane
2,2,6-trimethylheptane
2,2,5-trimethylheptane
2,2,3-trimethylheptane
2,2,4-trimethylheptane
2,2-dimethyl-3-ethylhexane
2,2-dimethyl-4-ethylhexane
2,5,5-trimethylheptane
2,3,3-trimethylheptane
2,4,4-trimethylheptane
3,3,5-trimethylheptane
3,3,4-trimethylheptane
3,4,4-trimethylheptane
3,3-dimethyl-4-ethylhexane
3,3-diethyl-2-methylhexane
2,3-dimethyl-3-ethylhexane
2,4-dimethyl-4-ethylhexane
3,4-dimethyl-3-ethylhexane
2,4-dimethyl-3-isopropylpentane
2,3,4,5-tetramethylhexane
2,2,4,5-tetramethylhexane
2,2,3,5-tetramethylhexane
2,2,4-trimethyl-3-ethylpentane
2,2,3,4-tetramethylhexane
2,2,3,5-tetramethylhexane
2,3,4,4-tetramethylhexane
2,3,3,4tetramethylhexane
2,3,4-trimethyl-3-ethylpentane
2,2,5,5-tetramethylhexane
2,2,4,4-tetramethylhexane
2,2,3,3-tetramethylhexane
2,2,3-trimethyl-3-ethylpentane
3,3,4,4-tetramethylhexane
2,2,3,4,4-~entamethylpentane
2,2,3,3,4-pentamethylpentane
W
Z
X
J
EAZ
150
149
138
141
145
151
146
143
143
142
131
134
134
138
141
137
138
135
129
133
130
125
146
138
134
121
126
129
136
135
124
127
131
130
128
122
123
125
139
134
130
131
122
126
131
127
127
126
123
122
118
114
119
122
117
117
121
124
123
115
118
120
116
115
112
127
119
115
110
111
79
80
81
83
84
65
69
71
68
67
72
70
73
72
74
75
71
73
77
76
76
80
60
66
64
76
69
72
61
56
63
62
64
63
65
67
68
70
51
55
57
52
58
56
57
59
53
59
62
61
64
68
63
60
68
54
58
47
48
50
53
49
4.808061
4.808 061
4.846 066
4.846066
4.846066
4.625 898
4.663 903
4.680739
4.663903
4.663903
4.718 744
4.701 908
4.718 744
4.701 908
4.701 908
4.718744
4.701 908
4.718 744
4.756750
4.739913
4.756749
4.794754
4.560660
4.621 321
4.621 321
4.742640
4.681 981
4.681 981
4.536582
4.519 745
4.591 423
4.574 586
4.574 586
4.574586
4.591 423
4.629428
4.629428
4.629427
4.416502
4.454507
4.481381
4.454507
4.519386
4.492512
4.477 163
4.504036
4.477 163
4.515 168
4.542041
4.542041
4.580046
4.625 356
4.564696
4.537823
4.602701
4.464 102
4.464 102
4.327 186
4.337223
4.392064
4.392064
4.359879
4.414 720
4.424757
4.447412
4.207 107
4.267 767
4.310660
4.371 320
4.371 320
4.154701
4.193 376
2.968 015
2.998419
3.295082
3.205535
3.086901
2.909 472
3.033297
3.129 602
3.124402
3.160036
3.499 857
3.390786
3.397 790
3.255 530
3.168 174
3.308840
3.268 555
3.375929
3.563688
3.412256
3.529 931
3.698 220
3.043 758
3.276962
3.417 512
3.874775
3.690295
3.575505
3.301403
3.337 430
3.728 003
3.603 336
3.461 674
3.502718
3.583 327
3.797908
3.756 108
3.685406
3.205456
3.355 508
3.518 426
3.469 465
3.808926
3.630845
3.464727
3.633 392
3.625 600
3.641 863
3.778 372
3.823 180
3.971 137
4.153 513
3.943 570
3.802 578
4.020494
3.983 500
3.813 995
3.684242
3.734823
4.072918
3.941 813
3.865559
4.034 119
4.089289
4.228 989
3.563001
3.887 595
4.101 784
4.328 342
4.281 757
4.231 133
4.403818
13.39022
14.03790
12.96797
13.560 17
13.57478
14.615 88
14.63300
14.64522
14.52239
14.43102
14.091 77
14.11981
14.08972
14.18572
15.25524
15.18455
14.398 18
14.39487
14.62580
14.731 24
13.877 44
13.99208
15.131 77
15.14273
14.94576
14.58404
14.94670
15.48928
15.76726
14.80487
15.08490
15.18979
15.66241
15.583 71
15.62695
14.98369
15.12535
16.322 97
15.651 22
16.31585
16.25524
15.381 43
15.66329
15.783 09
16.29656
16.23434
15.27246
16.129 13
16.14945
16.08077
15.585 07
14.99200
15.875 25
15.866 17
16.58203
15.27807
16.860 020
16.651 36
16.65644
15.96457
17.38966
16.56671
17.37724
17.36261
17.07437
17.381 50
17.09826
17.153 43
17.63852
17.89499
18.454 19
18.43252
111
108
55
56
57
41
43
47
52
53
40
43
is the degree of the ith vertex, and the V of 2-methylbutane
is
V = {1,3,2,1,1}
EAmax
2.412358
2.409867
2.231 204
2.235 614
2.242 122
2.659 508
2.646 8 12
2.779 339
2.663 637
2.720603
2.685 158
2.679002
2.688 596
2.650472
2.508 182
2.621908
2.573 861
2.615 580
2.506718
2.496 523
2.502991
2.371 022
3.902330
3.539 593
3.530 638
2.781 714
3.149601
3.159 644
2.797082
2.787 556
2.824479
2.772741
2.808 73 1
2.768 814
2.839 443
2.748271
2.797 341
2.729738
3.902401
3.902960
3.876 995
3.910422
3.854 864
3.907 570
3.541 770
3.560056
3.552846
3.553 431
3.525 855
3.518 440
3.494951
2.951 106
3.230296
3.206 202
3.172749
2.924988
2.924877
3.911 134
3.878260
3.859 398
3.878834
3.579 268
3.539 185
3.547729
3.287522
3.954520
3.976226
4.05223 1
3.949 285
3.821 292
4.020 850
4.050 615
bp ("C
165.7
165.1
162.0
163.64
166.0
159.87
158.54
164.31
156.8
153.0
160.0
160.0
166.0
159.7
160.0
166.0
160.0
162.1
167.0
158.3
167.0
162.0
155.0
161.2
157.5
166.3
167.0
163.8
155.7
144.8
163.0
157.0
157.0
157.0
163.0
164.0
164.0
164.0
148.2
148.0
158.0
147.7
159.0
147.0
152.8
160.1
153.0
155.68
164.0
164.0
165.0
174.0
169.0
158.0
170.0
157.04
161.0
148.2
148.4
155.3
154.9
153.0
162.2
164.59
169.44
137.46
153.3
158.0
168.0
170.5
159.29
166.05
The extended adjacency matrix (EA matrix) is represented
as follows:
J. Chem. Inf. Comput. Sci., Vol. 34, No. 5, 1994 1143
EXTENDED
ADJACENCY
MATRIXINDICES
Table 5. Results of Relationships of 149 Structures
141
143
142
indices
W
bp/OC
141
145
Z
x
146
147
148
149
J
I50
EA
a@&@ 4
152
151
153
. .
157
156
158
I59
160
,w,
+
+
X, = -2.657 551
9
/
3
I
4
10
11
S
0.9844
2281.71
12.08
0.9710
1202.20
16.81
0.9879
2951.82
10.00
0.9101
352.29
27.81
0.9914
4178.34
9.46
12
X , = 2.657 550 X , = 3.318 116 X 10"
X , = 1.108 644 X , =-1.108 644
EAC and EAmax are calculated respectively on the basis
of the definitions
&BW
2
+
+
149.99 0.14W1.19Wfz
bp/OC 138.23 0.55Z 2.42Z'f2
bp/'C = -5.13 - 3 . 1 4 ~
8.65~~
bp/'C = 78.59 - 36.91J
15.5152
b p / T = 134.19 + 3.45EAE
10.43EAmax
Figure 1. Skeletons of organic compounds showing different ring
structures including unusual ring systems.
1
F
constructed. EAC is the sum of the absolute eigenvalues of
the EA matrix, and EAmax is the maximum of the absolute
eigenvalues of the EA matrix. EAC and EAmax are together
called the EA indices. We also take 2-methylbutane as an
example for which the eigenvalues of its EA matrix are
15s
I54
R
ep
E A 1 = w,l+
wzl+ w31+ w,l+ w,l=
7.532390
EAmax = max(wA) = 2.657551
According to the preceding discussion, the hydrogensuppressed graph is a simple graph, and neither heteroatoms
nor multiple bonds were considered. Thus, the indices derived
from these graphs could not be expected to possessing high
selectivity; for example, 2-methylbutane, 2-butanol, and
3-methyl-1-butene have the same chemical graphs. Of course,
the EA indices are the same. In order to increase the
discriminating power of the new indices, two factors, heteroatoms and multiple bonds, embraced within species should
be subsequently taken into account.
1. Heteroatoms. The electronegativity is the quantitative
property of an atom, which is the power of an atom in a
molecule to attract electrons to itself and is concerned with
atoms in molecules. Therefore, we introduced the electronegativities in our topological indices so as to reflect the
influences of heteroatoms. Table 1 shows the electronegativities of common atoms.
If a compound contains heteroatoms, the degree vector and
the elements of the EA matrix will be changed as follows:
qJ-Ja+-+@-+
13
14
17
18
16
16
Figure 2. Skeletons of partial cyclic graphs with eight vertices and
the degree of each vertex being 4.
u: = uiELCi
(2)
whereas
&. = ELC,
(3)
where gij is an element of the EA matrix and ai, denotes an
element of matrix A. So, the extended adjacency matrix of
2-methylbutane is
0
1.67 0
0
0
1.67 0
1.08 1.67 0
where ELCi is the electronegativity of atom i; u: is used
instead of vi and is the new gii.
2. Multiple Bonds. The further extension is to enable
improved differentiation to better characterize structures
containing double and triple bonds.
u," = ui,
h
+ 'I,,
v," = v," + 'I3,
u," = v,"
1.67 0
Then, the eigenvalues of the EA matrix can be calculated,
and the new topological indices-EAC and EAmax-will be
b, = 1
u," = uj
+ 'I2,
v? = uj + 'I,,
b, = 2
bij = 3
where bij is the bond type to connect atoms i and j and u," is
1144 J . Chem. In& Comput. Sci., Vol. 34, No. 5, 1994
YANG ET AL.
Table 6. Two Topological Indices and Boiling Points of 37 Alcohols
no.
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
compound
methanol
ethanol
1-propanol
2-propanol
1-butanol
2-butanol
2-methyl- 1-propanol
2-methyl-2-propanol
1-pentanol
2-pentanol
3-pentaol
2-methyl- 1-butanol
3-methyl-1 -butanol
2-methyl-2-butanol
3-methyl-2-butanol
2,2-dimethyl- 1-propanol
1-hexanol
2-hexanol
3-hexanol
2-methyl- 1-pentanol
3-methyl- 1-pentanol
4-methyl- 1-pentanol
2-methyl-2-pentanol
3-methyl-2-pentanol
4-methyl-2-pentanol
2-methyl-3-pentanol
3-methyl-3-pentanol
2-ethyl-1 -butanol
2,2-dimethyl- 1-butanol
2,3-dimethyl- 1-butanol
3,3-dimethyl- 1-butanol
2,3-dimethyl-2-butanol
3,3-dimethyl-2-butanol
1-heptanol
1-octanol
1-nonanol
1-decanol
X
1X"
1.000 000
1.414 214
1.914 214
1.732 051
2.414 213
2.270 056
2.270 56
2.000 000
2.914 213
2.770 055
2.808 060
2.808 060
2.770 056
2.560 660
2.642 734
2.560 660
3.414 213
3.270 055
3.808 060
3.808 060
3.808 060
3.270 055
3.060 660
3.180 739
3.125 897
3.180 739
3.121 320
3.346 065
3.121 320
3.180 739
3.060 660
2.943 376
2.943 376
3.914 213
4.414 214
4.914 214
5.414 214
0.447 214
1.023 335
1.523 335
1.412 899
2.023 335
1.950 904
1.879 177
1.723 607
2.523 335
2.450 904
2.488 909
2.417 181
2.379 177
2.284 267
2.323 583
2.169 781
3.023 335
2.950 904
2.988 909
2.917 181
2.917 181
2.879 176
2.784 266
2.861 588
2.806 746
2.861 588
2.844 927
2.955 186
2.730 441
2.789 860
2.669 781
2.666 983
2.624 224
3.523 335
4.023 335
4.523 335
5.023 335
the new ui. Similar to eq 1, we have
vp/v;
g.. = a , ,
Y
Y
+ v;/vp
2
(4)
and the indices are expected to reflect the influences of
heteroatoms orland multiple bonds will be generated from
the modified EA matrix. For instance, the indices EAC and
EAmax tocompound 2-butanolare 13.500 00 and 5.149 825,
respectively.
EXAMINATION OF UNIQUENESS
The purpose of topological indices is to classify structures
and to serve for structure-property correlations. Nonuniqueness is not necessarily a disadvantage when correlation study
is the prime target, because there are compounds with similar
or the same properties which requires a similar or a same
index. However, interest continues in trying to devise an index
that would be unique. In this section we will exam the
selectivity of the new indices introduced in this study.
For examining the selectivitiesof the EA indices,over 20 000
structures and graphs have been detected. These structures
and graphs include the following families.
(1) Acyclic alkane molecular graphs up to n = 16 (n > 16,
not to be detected), the total number of alkane isomers being
18 030, can be differentiated without degeneracy by using the
EA indices.
(2) The structures and graphs selected by Randie* for
examining the uniqueness of ID numbers were detected,
containing monocyclic graphs up to n = 8 (1 12 cases), bicyclic
graphs up to n = 7 (79 cases), all graphs on five vertices,
sesquiterpenes (30 cases), and miscellaneous, etc. The EA
EAI:
6.000 000
8.500 000
1 1.000 00
11.209 88
13.500 00
13.500 00
13.748 93
16.399 06
16.000 00
16.000 00
16.000 00
16.000 00
16.275 47
18.161 19
16.536 68
18.799 45
18.500 00
18.500 00
18.500 00
18.500 00
18.500 00
18.780 81
20.643 11
18.699 99
18.927 43
18.856 79
19.876 70
18.500 00
20.570 58
19.047 25
21.304 26
20.695 65
21.257 00
2 1.ooo 00
23.500 00
26.000 00
28.500 00
EAmax
4.169 124
4.466 509
4.484 931
5.358 974
4.494 513
5.149 825
5.169 988
6.616 247
4.500 160
5.138 161
4.957 134
4.963 617
5.147 513
6.279 202
5.391 488
6.409 247
4.503 749
5.135 648
4.946 299
4.952 079
4.935 991
5.142 475
6.271 849
5.271 415
5.299 961
5.276 77
5.927 895
4.790 382
6.049 779
5.290 365
6.402 860
6.281 723
6.395 639
4.506 150
4.507 8 13
4.508 997
4.509 859
bp ("C)
64.7
78.3
97.2
82.3
117.7
99.6
107.9
82.4
137.8
119.0
115.3
128.7
131.2
102.0
111.5
113.1
157.7
139.9
135.4
148.0
152.4
151.8
121.4
134.2
131.7
126.5
122.4
146.5
136.8
149.0
143.0
118.6
120.0
176.3
195.2
213.1
230.2
indices show a remarkable ability to discriminate among the
structures and graphs. As an example, the EA index values
and the corresponding skeletons for graphs 141-160 as
numbered by Randie are shown in Table 2 and Figure 1,
respectively. Evidently, all of these structures can be
discriminated.
(3) For more rigorously determining the uniqueness of EA
indices, the cyclic graphs having n = 8 vertices, the degree of
each vertex being 4, offer novel comparisons. All of the 204
structures were enumerated with the program ISOM,9
developed for elucidation of structures of organic compounds
in our laboratory. The EA indices also show a high selectivity,
because most of the very closely related structures can be
discriminated by the EA indices, although there are some
pair counter examples having the same index values. As an
example, Figure 2 shows the partial cyclic graphs of these
structures, and Table 3 gives the corresponding values of E A 1
and EAmax. In which, 11 and 12,13 and 14,15 and 16, and
17 and 18 are the degeneracy pairs.
Note, that if two structures do not have the same EAC and
EAmax index values simultaneously, these two structures are
considered to be discriminated in this study.
APPLICATIONS TO CORRELATION
Topological indices developed for the purpose of obtaining
correlations with physicochemical properties and biological
activities of chemical substances have been applied for a very
extensive range. The current major applications include
bibliographical species classification, physicochemical parameter evaluation, and pharmaceutical drug design. In this
section the employed results will be given for alkanes, alcohols,
and barbiturates.
J. Chem. Znf. Comput. Sci., Vol. 34, No. 5, 1994 1145
EXTENDED
ADJACENCY
MATRIXINDICES
Table 7. Results of Relationships for 37 Alcohols
~
eq
R
F
S
bp/OC 29.64 + 88.831xv
bp/OC = -34.00 +
168.10'~- 127.98'~'
bp/OC = 132.14 + 7.70EAE26.36EAmax
0.9556
0.9923
1088.86
10.29
4.39
0.9838
511.90
6.35
indices
Ixv
lxlx'
EA
Table 8. log P and the EA Indices for Barbiturates with Structure I
R1
no.
1
2
3
4
5
6
7
R2
ethyl
methyl
ethyl
ethyl
propyl
ethyl
isopropyl ethyl
methyl
methyl
ethyl
methyl
propyl
methyl
8 isopropyl methyl
propyl
9 methyl
propyl
10 ethyl
11 methyl
isopropyl
butyl
12 methyl
13 ethyl
butyl
14 ethyl
ethyl
Table 9.
no.
I5
16
17
18
19
20
21
22
23
24
25
1011P
R3
EAE
EAmax
logP
methyl
methyl
methyl
methyl
ethyl
ethyl
ethyl
ethyl
methyl
methyl
methyl
methyl
methyl
propyl
42.442 200
44.295 792
46.774 139
47.172211
42.610088
44.549950
47.032730
46.949 421
44.936 668
46.786049
45.041 950
47.435 841
49.283 989
49.262020
5.588 789
5.355 969
5.349 703
5.443 005
5.647092
5.415 510
5.406 142
5.412 628
5.587068
5.354 176
5.610 556
5.586836
5.353 878
5.350 319
1.15
1.65
2.15
1.95
1.15
1.65
2.15
1.95
1.65
2.15
1.45
2.15
2.65
2.65
It is clear that the better result can be obtained with EA
indices, though the results obtained using x and lxVis slightly
superior to ours.
3. Barbiturates. Barbiturates were thought to be nonspecific narcotic agents principally because log P (P = partition
coefficient in octanol-water) correlates very well with their
biological potency.ll Other studies1*showed a dependence of
the action of barbiturates upon chemical structure. Therefore,
it was of interest to carry out a correlation analysis of log P
and topological parameters. Correlations between the EA
indices and log P for barbiturates have been revealed by this
investigation.
The EA indices and log P for 25 barbiturate acid derivatives
with structure I and structure I1 are list in Table 8 and Table
9, respectively.
ox=;H
R2
0
NKN
0
R2
methyl
ethyl
propyl
butyl
methyl
ethyl
propyl
butyl
isobutyl
amyl
isoamyl
1-methyl, lapropenyl
1-methyl, 1-propenyl
1methyl, 1-propenyl
1-methyl, 1-propenyl
1-methylvinyl
1methylvinyl
1methylvinyl
1methylvinyl
1-methylvinyl
1methylvinyl
1methylvinyl
EAE
40.068 581
41.963 631
44.444 069
46.940 262
37.686 401
39.664 139
42.148 720
44.646 042
44.718 639
47.145 580
47.212 662
EAmax
log P
-
5.633 902 0.65
5.396 735 1.15
5.390 218 1.65
5.388 160 2.15
5.675 070 0.15
5.447 943 0.65
5.441 714 1.15
5.440 750 1.65
5.461 265 1.45
5.440 601 2.15
5.459 260 1.95
1. Alkanes. It is of interest to test methods on data for
alkanes because good data are generally available for complete
isomer sets. The first 150alkanes (Le., up to 10carbon atoms)
have been used by Mihalic et al.1° for a comparison of
performances of the 10 distance indices and 2 connectivity
indices in the structure-boiling point correlations. In this
paper, the same set of compounds except methane, Le., 149
alkanes, were employed, and EA indices of all of the 149
alkanes were calculated and are listed in Table 4. For
comparisons, the index values calculated by other schemes,
W,' Z,2x , and
~ J4 are also given in Table 4.
Table 5 shows a summary of the correlation coefficients,
r, standard deviations, s, and F-test values by all of the indices.
Evidently, the best result has been achieved by using EA
indices.
2. Alcohols. Because the most interesting structures
possessing activity are rather complex molecules with multiple
bonds and/or heteroatoms, it is quite important that topological
indices are able to correlate these kinds of molecules. Alcohol
containsone heteroatom, oxygen, whose ELC is 3.5 (see Table
1).
Similarly, for comparisons, besides boiling points of 37
alcohols and their EA indices, x and lxV are also given in
Table 6. The results of the correlation analysis are shown in
Table 7 .
NKN
0
II
I
and the EA Indices for Barbiturates with Structure I1
R1
/R3
The results of thecorrelation analysis between the EA indices
and log P values of these compounds are
log P = -2.8302
R = 0.9910
+ 0.1900EAC - 0.7395EAmax
F = 602.8965
S = 0.0861
n = 25
ACKNOWLEDGMENT
The financial support of the Natural Science Foundation
of China is gratefully acknowledged.
REFERENCES AND NOTES
(1) Wiener, H. Structural Determination of Paraffin Boiling Points. J.
Am. Chem. Soc. 1947,69, 17-20.
(2) Hosoya, H. Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated
Hydrocarbons. Bull. Chem. Soc. Jan. 1971,44, 2332-2339.
(3) RandiC, M. On CharacterizationofMolecular Branching. J. Am. Chem.
Soc. 1975, 97,66094615.
(4) Balaban, A. T. Topological Indices Based on Topological Distances in
Molecular Graphs. Pure Appl. Chem. 1983,55, 199-206.
(5) Kier, I. B.;Hall, L. H. Molecular Connectivity in Chemistry and Drug
Research; Academic: New York, 1976.
(6) RandiC, M.; Trinajstic, N. In search for graph invariants of chemical
interest. J. Mol. Srruct. (THEOCHEM) 1993, 300, 551-571.
(7) RandiC, M.; Trinajstic,N. Viewpoint 4-Comparative structure-property
studies: the connectivity basis. J. Mol. Strucr. (THEOCHEM) 1993,
284,209-221.
( 8 ) RandiC, M. On Molecular Identification Numbers. J . Chem. Inf.
Comput. Sci. 1984, 24, 164175.
(9) Hu, C.-Y.; Xu, L. Studies on Expert System for the Elucidation of the
Structures of Organic Compounds-StructuralGenerator of ESESOC11. Sei. China, in press.
(10) Mihalic, Z.; Nikolic, S.; Trinajstic, N. ComparativeStudy of Molecular
Descriptors Derived from the Distance Matrix. J. Chem. Inf. Compur.
Sei. 1992, 32, 28-37.
(1 1) Hansch, C.; Anderson, S. M.Structure-Activity Relation in Barbiturates
and Its Similarity to That in Other Narcotics. J . Med. Chem. 1967,10,
745-153.
(12) Bask, S. C.; Monsrud, L. J.; Rosen, M. E.; Frane, C. M.; Macnuson,
V. R.A Comparative Study of Lipophilicity and Topology Indexes in
Biological Correlation. Acra Pharm. Jugosl. 1986, 36, 8 1-95.