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Introduction
The molecular descriptor is the final result of logic and mathematical procedure which transform chemical information encoded within a symbolic representation of a molecule into a useful member or the result of some standardized experiments. Attention is paid to the term “useful” with its double meanings. It means that the number can give more insights into the interpretation of the molecular properties and / or is able to take part in a model for the prediction of some interesting property of the molecules.
The numerical invariants of chemical graphs are increasingly being used for a single number characterization of the corresponding chemical compounds [5]. These invariants are named in the chemical literature as topological indices [1, 2] or graph-theocratical indices [29]. The former term is the more common of the two. Topological indices have found application in various areas of chemistry, physics, mathematics, informatics, biology, etc [29], but their most important use to date is in the non-empirical Quantitative Structure- Property Relationships (QSPR) and Quantitative Structure -Activity Relationships (QSAR) [4, 24, 26, 27].
Survey on Graph Theocratical Matrices
Vertex Adjacency Matrix: The term vertex adjacency matrix was first introduced in chemical graph theory by Mallion in his interesting paper [20] on graph theocratical aspects of the ring current theory. Below we give the vertex adjacency matrix of the vertex labeled graph G.
et G = (V,E) be a grapLh where V = {1,2,3,··· ,n} the vertex adjacency matrix of a graph with vertex set V = {1,2,3,··· ,n} is the n × n matrix in which aij = 1 if and only if there is an adjacency from vertex i to vertex j. Each diagonal entry in the adjacency matrix of a graph is zero. i.e.,
Forgotten Adjacency Matrix: Recently, Gutman and Furtula [7] have studied the forgotten topological index F(G) of a molecular graph G. Based on the definition of adjacency matrix and vertex degrees of a graph G, we define the forgotten adjacency matrix as follows:
let G = (V,E) be a graph where V = {1,2,3,···, n} then the forgotten adjacency matrix F(G) of a graph G is defined as follows i.e.,
Degree-Sum Matrix: Ramane et. al [22] have introduced degree-sum matrix associated with a graph and obtained some upper and lower bounds for its eigenvalues.
let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the degree-sum matrix H(G) of a graph G is defined as follows i.e.,
Laplacian Matrix: In [11] Gutman and Zhou have put forward the Laplacian matrix. The Laplacian matrix sometimes also called a Kirchoff matrix [3] due to its role in matrix tree theorem, Implicitin the electrical network work of Kirchoff in his paper Kirchoff also introduced the concept of the spanning tree.
let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the Laplacian matrix L(G) of a graph G is defined as follows i.e.,
$$\begin{array}{}
\displaystyle
{a_{ij}} = \left\{ \begin{array}{*{20}{l}}
{ - 1,}&{{\text{if}\;}{v_i} \ne {v_j}{\rm{ and }}{v_i}{\;\text{is adjacent to}\;}{v_j};}\\
0&{{\text{if}\;}{v_i} \ne {v_j}{\rm{ and }}{v_i}{\rm{ is not adjacent to }}{v_j};}\\
{deg{v_i},}&{{v_i} = {v_j}.}
\end{array}\right.
\end{array}$$
Sum-Connectivity Matrix: The sum-connectivity matrix denoted by SCij was introduced independently by Zhou and Trianjstic [34] it is defined as follows:
let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the harmonic matrix SCij(G) of a graph G is defined as follows i.e.,
Vertex Randić Matrix: The vertex-connectivity matrix denoted by Rij introduced by Randic [23]. It can be regarded as edge-weighted matrix of the graph defined as:
let G = (V,E) be a graph where V = {1,2,3,··· ,n} then the harmonic matrix R(G) of a graph G is defined as follows i.e.,
The Use of Graph Theoretical Matrices in QSPR Studies
We have used nine Graph Theoretical Matrices viz, vertex-adjacency matrix, vertex Zagreb adjacency Matrix, forgotten adjacency matrix, harmonic matrix, geometric-airthmetic matrix, degree-sum matrix, laplacian matrix, sum-connectivity matrix and vertex Randi ć matrix respectively for modeling eight representative physical properties [boiling points(BP), molar volumes (mv) at 20°C, molar refractions (mr) at 20°C, heats of vaporization (hv) at 25°C, surface tensions (st) 20°C and melting points (mp)] of the 68 alkanes from n-butanes to nonanes. Values for these property were taken from Dejan Plavsi ć et. al [21]. The corresponding energy of the above said matrices and the experimental values for the physical properties of 68 alkanes are listed in Table 1 and 2 respectively.
S.No.
Alkane
bp(°C)
mv(cm3)
mr(cm3)
hv(kJ)
ct(°C)
cp(atm)
st(dyne/cm)
mp(°C)
1
Butane
-0.500
152.01
37.47
-138.35
2
2-methyl propane
-11.730
134.98
36
-159.60
3
Pentane
36.074
115.205
25.2656
26.42
196.62
33.31
16.00
-129.72
4
2-methyl butane
27.852
116.426
25.2923
24.59
187.70
32.9
15.00
-159.90
5
2,2 dimethylpropane
9.503
112.074
25.7243
21.78
160.60
31.57
-16.55
6
Hexane
68.740
130.688
29.9066
31.55
234.70
29.92
18.42
-95.35
7
2-methylpentane
60.271
131.933
29.9459
29.86
224.90
29.95
17.38
-153.67
8
3 -methyalpentane
63.282
129.717
29.8016
30.27
231.20
30.83
18.12
-118.00
9
2,2-methylbutane
49.741
132.744
29.9347
27.69
216.20
30.67
16.30
-99.87
10
2,3 -dimethylbutane
57.988
130.240
29.8104
29.12
227.10
30.99
17.37
-128.54
11
Heptanes
98.427
146.540
34.5504
36.55
267.55
27.01
20.26
-90.61
12
2-methylhexane
90.052
147.656
34.5908
34.80
257.90
27.2
19.29
-118.28
13
3-methylhexane
91.850
145.821
34.4597
35.08
262.40
28.1
19.79
-119.40
14
3-ethylpentane
93.475
143.517
34.2827
35.22
267.60
28.6
20.44
-118.60
15
2,2-dimethylpentane
79.197
148.695
34.6166
32.43
247.70
28.4
18.02
-123.81
16
2,3 -dimethylpentane
89.784
144.153
34.3237
34.24
264.60
29.2
19.96
-119.10
17
2,4-dimethylpentane
80.500
148.949
34.6192
32.88
247.10
27.4
18.15
-119.24
18
3,3-dimethylpentane
86.064
144.530
34.3323
33.02
263.00
30
19.59
-134.46
19
Octane
125.665
162.592
39.1922
41.48
296.20
24.64
21.76
-56.79
20
2-methylheptane
117.647
163.663
39.2316
39.68
288.00
24.8
20.60
-109.04
21
3-methylheptane
118.925
161.832
39.1001
39.83
292.00
25.6
21.17
-120.50
22
4-methylheptane
117.709
162.105
39.1174
39.67
290.00
25.6
21.00
-120.95
23
3-ethylhexane
118.53
160.07
38.94
39.40
292.00
25.74
21.51
24
2,2-dimethylhexane
10.84
164.28
39.25
37.29
279.00
25.6
19.60
-121.18
25
2,3-dimethylhexane
115.607
160.39
38.98
38.79
293.00
26.6
20.99
26
2,4-dimethylhexane
109.42
163.09
39.13
37.76
282.00
25.8
20.05
-137.50
27
2,5-dimethylhexane
109.10
164.69
39.25
37.86
279.00
25
19.73
-91.20
28
3,3-dimethy lhexane
111.96
160.87
39.00
37.93
290.84
27.2
20.63
-126.10
29
3,4-dimethy lhexane
117.72
158.81
38.84
39.02
298.00
27.4
21.64
30
3 -ethyl-2-methylpentane
115.65
158.79
38.83
38.52
295.00
27.4
21.52
-114.96
31
3 -ethyl-3 -methylpentane
118.25
157.02
38.71
37.99
305.00
28.9
21.99
-90.87
32
2,2,3-trimethylpentane
109.84
159.52
38.92
36.91
294.00
28.2
20.67
-112.27
33
2,2,4-trimethylpentane
99.23
165.08
39.26
35.13
271.15
25.5
18.77
-107.38
34
2,3,3-trimethylpentane
114.76
157.29
38.76
37.22
303.00
29
21.56
-100.70
35
2,3,4-trimethylpentane
113.46
158.85
38.86
37.61
295.00
27.6
21.14
-109.21
36
2,2, 3,3-tetramethylbutane
106.470
270.8
24.5
36
Nonane
150.79
178.71
43.84
46.44
322.00
22.74
22.92
-53.52
37
2-methyloctane
143.26
179.77
43.87
44.65
315.00
23.6
21.88
-80.40
38
3-methyloctane
144.18
177.95
43.72
44.75
318.00
23.7
22.34
-107.64
39
4-methyloctane
142.48
178.15
43.76
44.75
318.30
23.06
22.34
-113.20
40
3-ethylheptane
143.00
176.41
43.64
44.81
318.00
23.98
22.81
-114.90
41
4-ethylheptane
141.20
175.68
43.49
44.81
318.30
23.98
22.81
42
2,2-dimethylheptane
132.69
180.50
43.91
42.28
302.00
22.8
20.80
-113.00
43
2,3-dimethylheptane
140.50
176.65
43.63
43.79
315.00
23.79
22.34
-116.00
44
2,4-dimethylheptane
133.50
179.12
43.73
42.87
306.00
22.7
23.30
45
2,5-dimethylheptane
136.00
179.37
43.84
43.87
307.80
22.7
21.30
46
2,6- dimethylheptane
135.21
180.91
43.92
42.82
306.00
23.7
20.83
-102.90
47
3,3- dimethylheptane
137.300
176.897
43.6870
42.66
314.00
24.19
22.01
48
3,4- dimethylheptane
140.600
175.349
43.5473
43.84
322.70
24.77
22.80
49
3,5- dimethylheptane
136.000
177.386
43.6379
42.98
312.30
23.59
21.77
50
4,4- dimethylheptane
135.200
176.897
43.6022
42.66
317.80
24.18
22.01
51
3-ethyl-2-methylhexane
138.000
175.445
43.6550
43.84
322.70
24.77
22.80
52
4-ethyl-2-methylhexane
133.800
177.386
43.6472
42.98
330.30
25.56
21.77
53
3-ethyl-3-methylhexane
140.600
173.077
43.2680
44.04
327.20
25.66
23.22
54
2,2,4- trimethylhexane
126.540
179.220
43.7638
40.57
301.00
23.39
20.51
-120.00
55
2,2,5- trimethylhexane
124.084
181.346
43.9356
40.17
296.60
22.41
20.04
-105.78
56
2,3,3- trimethylhexane
137.680
173.780
43.4347
42.23
326.10
25.56
22.41
-116.80
57
2,3,4- trimethylhexane
139.000
173.498
43.4917
42.93
324.20
25.46
22.80
58
2,3,5- trimethylhexane
131.340
177.656
43.6474
41.42
309.40
23.49
21.27
-127.80
59
3,3,4- trimethylhexane
140.460
172.055
43.3407
42.28
330.60
26.45
23.27
-101.20
60
3,3-diethylpentane
146.168
170.185
43.1134
43.36
342.80
26.94
23.75
-33.11
61
2,2-dimethyl-3-ethylpentane
133.830
174.537
43.4571
42.02
322.60
25.96
22.38
-99.20
62
2,3-dimethyl-3-ethylpentane
142.000
170.093
42.9542
42.55
338.60
26.94
23.87
63
2,4-dimethyl-3-ethylpentane
136.730
173.804
43.4037
42.93
324.20
25.46
22.80
-122.20
64
2,2,3,3-tetramethylpentane
140.274
169.495
43.2147
41.00
334.50
27.04
23.38
-99.0
65
2,2,3,4- tetramethylpentane
133.016
173.557
43.4359
41.00
319.60
25.66
21.98
-121.09
66
2,2,4,4- tetramethylpentane
122.284
178.256
43.8747
38.10
301.60
24.58
20.37
-66.54
67
2,3,3,4- tetramethylpentane
141.551
169.928
43.2016
41.75
334.50
26.85
23.31
-102.12
S.No.
Alkane
E(G)
Z1E(G)
FE(G)
HE(G)
GAE(G)
DSE(G)
LE(G)
SCE(G)
RE(G)
1
Butane
2.828
10
18
2.844
4.268
18.166
6
2.516
3
2
2-methyl propane
2.828
11.243
30
1.732
3
18.422
6
1.732
2
3
Pentane
4.472
14.001
25.999
3.274
5.286
25.874
8
2.98
3.414
4
2-methyl butane
5.226
16
38
2.86
4.758
26.253
8
2.72
3.154
5
2,2 dimethylpropane
4
20
68
1.6
3.2
26.88
8
1.788
2
6
Hexane
6.988
17.99
34
4.086
6.788
33.698
10
3.768
4.236
7
2-methylpentane
6.064
14.999
46
3.254
5.722
34.168
10
3.15
3.528
8
3 -methyalpentane
6.9
20
46
3.924
6.442
34.168
10
3.652
4.23
9
2,2-methylbutane
5.818
23.992
76.001
2.798
5.024
34.485
10
2.824
3.224
10
2,3 -dimethylbutane
6.004
22
58
2.906
5.292
34.628
10
2.944
3.334
11
Heptanes
8.054
21.999
41.98
4.59
7.868
41.58
12
4.284
4.732
12
2-methylhexane
7.728
24.001
53.991
4.282
7.266
42.12
11.999
4.17
4.376
13
3-methylhexane
7.88
24
54
4.376
7.456
43.058
11.999
4.112
4.632
15
3-ethylpentane
6.9
20
56
4.586
7.654
38.08
10
4.24
4.828
16
2,2-dimethylpentane
6.72
27.999
83.999
3.176
5.966
43.06
12.001
3.24
3.582
17
2,3 -dimethylpentane
7.664
24.999
65.999
3.966
6.92
42.648
12.001
3.872
4.404
18
2,4-dimethylpentane
6.156
20
46
3.226
6.148
43.015
10
3.312
3.632
19
3,3-dimethylpentane
6.596
28
83.999
3.944
6.802
43.266
12
3.666
5.74
20
Octane
9.516
26
50
5.324
9.312
49.496
14
4.76
5.468
21
2-methylheptane
8.764
28
62
4.792
8.294
50.09
14
4.33
4.82
22
3-methylheptane
9.408
27.999
62
5.138
8.924
49.996
14
4.608
5.41
23
4-methylheptane
8.828
27.999
62
4.734
8.402
50.09
14
4.298
4.974
24
3-ethylhexane
7.88
24
54
5.282
9.034
50.09
11.999
4.536
5.502
25
2,2-dimethylhexane
8.312
31.999
91.999
4.008
7.522
51.14
14
3.892
4.424
26
2,3-dimethylhexane
8.646
30.001
73.999
4.376
7.95
50.671
15
4.198
4.792
27
2,4-dimethylhexane
8.564
30
74
4.314
7.862
49.825
14.001
4.114
4.678
28
2,5-dimethylhexane
8.472
30
74
3.714
7.095
50.671
14
3.761
4.468
29
3,3-dimethylhexane
8.52
31.998
92
4.334
7.772
51.14
14
3.968
4.752
30
3,4-dimethylhexane
9.332
30
74
5.002
8.584
50.413
14.001
4.5
5.41
31
3-ethyl-2-methylpentane
7.664
29
70
4.588
4.51
51.016
14
4.16
4.916
32
3-ethyl-3-methylpentane
7.596
32
91.246
5.048
8.526
51.14
14
4.336
5.488
33
2,2,3 -trimethylpentane
7.3
34.001
104
3.902
7.228
51.876
16
3.808
4.448
34
2,2,4-trimethylpentane
7.384
33.999
104
3.1444
6.386
51.701
14
3.056
3.684
35
2,3,3-trimethylpentane
8.054
34
104.001
4.026
7.374
47.686
14
4.068
4.6
36
2,3,4-trimethylpentane
8.424
32
86.001
4.002
7.5
51.24
14
4.09
4.574
36
2,2,3,3-tetramethylbutane
7.212
38
134
2.816
5.892
52.698
15.998
3.179
3.5
37
Nonane
10.628
30
58.001
5.884
10.432
57.432
16
5.574
6.028
38
2-methyloctane
10.252
32
70.001
5.342
9.792
58.07
16
5.22
5.22
39
3-methyloctane
10.472
32
69.998
5.662
10.042
58.07
15.999
5.418
5.954
40
4-methyloctane
10.384
32
70.001
5.58
9.97
58.07
16
5.354
5.858
41
3-ethylheptane
10.564
28.126
69.999
5.864
10.214
58.07
15.999
5.039
5.49
42
4-ethylheptane
10.492
32
70
5.79
10.138
57.842
16
5.3
5.902
43
2,2-dimethylheptane
9.336
35.999
99.999
4.502
8.754
59.21
16
4.355
4.916
44
2,3 -dimethylheptane
10.176
34.001
81.999
5.202
9.49
58.694
16
5.118
5.632
45
2,4-dimethylheptane
9.508
34
81.999
4.728
8.866
58.694
16
4.718
5.132
46
2,5 -dimethylheptane
10.152
34
81.999
5.162
9.438
58.694
16
5.092
5.598
47
2,6- dimethylheptane
10.096
35.999
99.999
4.564
8.536
59.21
15.999
4.64
4.98
48
3,3- dimethylheptane
9.464
33.999
82
5.194
9.324
58.694
16
5.072
5.632
49
3,4- dimethylheptane
10.312
34
81.999
5.45
9.672
59.079
15.999
5.262
3.939
50
3,5- dimethylheptane
10.29
33.999
82.001
5.418
9.628
58.978
16
5.24
5.85
51
4,4- dimethylheptane
9.43
36.001
99.999
4.744
8.73
59.588
16
4.684
5.146
52
3-ethyl-2-methylhexane
10.198
33.999
81.999
5.35
9.606
58.694
16
5.184
5.73
53
4-ethyl-2-methylhexane
10.176
34.001
81.999
5.308
9.55
58.694
15.91
5.158
5.698
54
3-ethyl-3-methylhexane
10.262
36
99.999
5.504
9.56
59.21
15
5.252
5.952
55
2,2,4- trimethylhexane
9.13
38
111.999
4.242
8.134
59.814
15.999
4.372
4.796
56
2,2,5- trimethylhexane
9.06
37.993
112
4.022
8.012
59.814
16
4.256
4.582
57
2,3,3- trimethylhexane
9.3
37.999
112
4.418
8.34
59.814
16.176
4.496
4.97
58
2,3,4- trimethylhexane
10.096
36
93.999
5.068
9.184
59.307
16.001
5.02
5.648
59
2,3,5- trimethylhexane
9.336
36.017
94
4.366
8.428
59.307
15.999
4.5
4.918
60
3,3,4- trimethylhexane
10.036
44
112.002
5.086
9.052
59.814
16
4.994
5.67
61
3,3-diethylpentane
10.472
36
100
5.793
7.736
59.21
15.55
4.903
3.042
62
2,2-dimethyl-3-ethylpentane
9.3
37.999
59.814
4.528
6.954
112.003
16
4.476
4.498
63
2,3-dimethyl-3-ethylpentane
10.062
38.001
61.877
5.116
8.43
112.001
16.001
3.976
4.376
64
2,4-dimethyl-3-ethylpentane
8.884
31
57.368
5.088
8.78
74.999
14
4.818
5.582
65
2,2,3,3-tetramethylpentane
8.98
42
60.9
4.24
7.35
142
15.998
4.122
4.672
66
2,2,3,4- tetramethylpentane
9.02
40
60.408
3.93
7.75
124
16.001
4.168
4.614
67
2,2,4,4- tetramethylpentane
7.936
42
60.9
3.056
6.61
142
16
4.44
3.726
68
2,3,3,4- tetramethylpentane
9.152
39.243
57.278
4.14
7.96
124
14.999
4.992
4.74
Regression Models
We have tested the following linear regression model
$$\begin{array}{}
\displaystyle
P = A + B\left( {TI} \right)
\end{array}$$
where P = physical property, TI = topological index.
Using (3.1), we have obtained the following different linear models for each degree based topological index, which are listed below.
Vertex adjacency energyE(G):
$$\begin{array}{}
\displaystyle
bp = - 51.397 + \left[ {E\left( G \right)} \right]19.268
\end{array}$$
Statical parameters for the linear QSPR model for E(G).
Physical Properties
N
a
b
r
s
F
Boiling point
68
-51.397
19.268
0.959
10.54257
763.634
Molar volume
65
75.5727
10.1894
0.905
7.62675
286.565
Molar refraction
65
13.1325
3.0530
0.912
2.19071
311.823
Heats of vaporization
65
10.0231
3.3206
0.968
1.3755
935.676
Critical temperature
68
91.9103
23.1023
0.936
16.0513
496.344
Critical Pressure
68
28.9043
0.0858
0.006
565.769
0.003
Surface tension
64
11.0052
1.1474
0.871
0.9705
194.411
Melting point
52
-145.3088
4.4911
0.316
25.8886
5.528
Statical parameters for the linear QSPR model for Z1E(G).
Physical Properties
N
a
b
r
s
F
Boiling point
68
-13.358
4.1092
0.837
20.46726
154.120
Molar volume
65
94.714914
2.201775
0.843
9.6547
155.135
Molar refraction
65
18.6119
0.6778
o.873
2.6070
201.663
Heats of vaporization
65
14.0714
0.7822
0.736
3.7075
74.461
Critical temperature
68
211.4016
1.0167
0.855
23.6776
180.025
Critical Pressure
68
32.606
-0.0988
0.030
119.570
5.061
Surface tension
64
14.4576
0.2108
0.727
1.35607
69.337
Melting point
52
-137.1762
6.6745
0.312
25.9224
5.383
Statical parameters for the linear QSPR model for FE(G).
Physical Properties
N
a
b
r
s
F
Boiling point
68
49.581
0.829
0.524
31.8288
25.020
Molar volume
65
127.4698
0.4683
0.551
14.9892
27.500
Molar refraction
65
29.0759
0.1390
0.55
4.4616
27.365
Heats of vaporization
65
490.0581
-5.8696
0.410
4.9948
12.738
Critical temperature
68
131.1926
5.1377
0.525
38.8983
25.158
Critical Pressure
68
31.4563
-0.0249
0.024
135.575
0.038
Surface tension
64
18.4178
0.0341
0.369
1.83471
9.750
Melting point
52
-123.9450
0.2094
0.187
26.7987
1.821
Statical parameters for the linear QSPR model for HE(G).
Physical Properties
N
a
b
r
s
F
Boiling point
68
-29.71
32.08
0.825
21.1104
140.91
Molar volume
65
100.054
13.9260
0.682
13.1375
54.809
Molar refraction
65
20.4799
4.2371
0.698
3.8278
59.766
Heats of vaporization
65
31.9676
1.4827
0.856
2.8351
172.070
Critical temperature
67
118.2041
38.4066
0.808
26.9494
123.915
Critical Pressure
68
27.5611
0.4722
0.018
24.6671
0.022
Surface tension
64
12.1028
1.9642
0.804
1.1734
113.396
Melting point
52
-137.1762
6.6745
0.239
26.4944
3.018
Statical parameters for the linear QSPR model for GAE(G).
Physical Properties
N
a
b
r
s
F
Boiling point
68
-36.99
18.984
0.870
18.4148
205.922
Molar volume
65
90.0410
9.0967
0.791
11.0019
104.98
Molar refraction
65
18.0357
2.6920
0.787
3.2997
102.211
Heats of vaporization
65
34.6289
0.5068
0.89
2.4722
246.160
Critical temperature
68
113.4305
22.2200
0.833
25.3083
149.343
Critical Pressure
68
27.3753
0.2895
0.020
24.661
0.027
Surface tension
64
12.6092
1.0425
0.767
1.2654
88.825
Melting point
52
-132.52
3.28
0.208
26.6869
2.256
Statical parameters for the linear QSPR model for DSE(G).
Physical Properties
N
a
b
r
s
F
Boiling point
68
59.931
0.897
0.597
29.9848
36.560
Molar volume
65
138.935
0.4028
0.549
15.0099
27.250
Molar refraction
65
31.1877
0.1300
0.596
4.2888
34.79
Heats of vaporization
65
32.5560
0.1074
0.481
4.8027
18.918
Critical temperature
68
219.9439
1.1727
0.638
35.1156
45.347
Critical Pressure
68
32.1358
-0.0444
0.045
81.216
0.133
Surface tension
64
18.4122
0.0436
0.540
1.6617
25.459
Melting point
52
-122.628
0.2430
0.244
26.4571
3.164
Statical parameters for the linear QSPR model for LE (G).
Physical Properties
N
a
b
r
s
F
Boiling point
68
105.998
0.795
0.951
11.6074
618.389
Molar volume
65
61.7923
7.1569
0.964
4.7915
822.621
Molar refraction
65
9.2778
2.1462
0.972
1.2624
1065.626
Heats of vaporization
65
28.1626
0.7323
0.924
2.0990
365.873
Critical temperature
68
71.1141
15.5601
0.935
16.1580
462.301
Critical Pressure
68
31.3140
-0.1221
0.014
24.6698
0.012
Surface tension
64
10.9055
0.7551
0.834
1.0888
141.725
Melting point
52
-146.7815
2.8215
0.301
26.0151
4.989
Statical parameters for the linear QSPR model for SCE(G).
Physical Properties
N
a
b
r
s
F
Boiling point
68
-57.214
39.635
0.907
15.7086
307.683
Molar volume
65
79.5397
19.1531
0.823
10.1925
132.724
Molar refraction
65
14.5174
5.7627
0.833
2.9563
142.822
Heats of vaporization
65
32.3464
1.44459
0.931
1.9928
412.801
Critical temperature
68
86.5805
47.321
0.883
21.4546
233.650
Critical Pressure
68
26.5275
0.7319
0.025
24.6642
0.043
Surface tension
64
0.7399
2.3432
0.847
1.0489
157.493
Melting point
52
-144.7607
8.7341
0.278
26.2084
4.181
Statical parameters for the linear QSPR model for R(G).
Physical Properties
N
a
b
r
s
F
Boiling point
68
-33.08
30.673
0.776
23.5965
99.608
Molar volume
65
4740.5335
-960.1500
0.655
13.5743
47.350
Molar refraction
65
20.0386
4.0559
0.656
4.0335
47.567
Heats of vaporization
65
24.5656
2.8952
0.764
3.53619
88.107
Critical temperature
68
116.5371
36.2135
0.749
30.3071
84.165
Critical Pressure
68
25.3061
0.9224
0.035
29.00
0.083
Surface tension
64
13.0761
1.6338
0.667
1.46987
49.788
Melting point
52
-113.7685
1.0137
0.035
27.2655
0.061
Discussion and Concluding Remarks
By inspection of the data in Tables 3 to 11, it is possible to draw a number of conclusions for the given energy like invariants.
First, the famous and much studied invariant, energy of a graph found more suitable tool to predict physical property of alkane, especially Boiling points, Molar volume, Surface tension, Critical temperature, Heats of vaporization and Molar refractions of alkanes with correlation coefficient values r = 0.959,0.905,0.871,0.968 and 0.912 respectively.
Motivated by vertex Zagreb energy. Here we introduced a new topological invariant namely, vertex Zagreb adjacency energy. The QSPR study of vertex Zagreb energy reveals that Z1E(G) can be useful in predicting the Boiling points, Critical temperatures, Molar volumes and Molar refraction of alkanes also from Table 4,we can see that the correlation coefficient value of Z1E(G) with physical properties of alkanes lies between 0.030 to 0.873.
In addition by using the recently advocated idea of using Forgotten index in QSPR studies, we introduced Forgotten adjacency energy. The QSPR study of FE(G) shows that the idea of using FE(G) in QSPR study does not make sense. Since the correlation coefficient values FE(G) with physical properties of alkanes lies between 0.024 to 0.551.
The harmonic index did not attract anybody’s attention, especially, not of chemists. No chemical applications of the harmonic index were reported so far, but knowing the present situation in the mathematical chemistry. We here explore the chemical applications of harmonic index. The Table 6 reveals that harmonic energy is also useful tool in predicting the Boiling point, Heats of vaporization, Surface tensions and Critical temperature of alkanes with correlation coefficient values r = 0.825,0.856,0.804 and 0.808, respectively.
The QSPR study of Geometric-arithmetic energy reveals that the predicting power of G − A- energy for the physical properties Boiling points, Heats of vaporization and Critical temperatures of alkanes with correlation coefficient values r = 0.870,0.89 and 0.833 respectively.
In addition the results for degree sum energy revealed that the recent advocated idea of using degree sum energy doesn’t pass the test.
The so called Laplacian energy shows remarkably good correlation with the Boiling points, Molar volumes, Molar refractions, Heats of vaporization, Surface tensions and Critical temperatures of alkanes with correlation coefficient values r = 0.951,0.964,0.972,0.924,0.83 and r = 0.935 respectively. Further, the correlation coefficient values lies between 0.014 to 0.972. In fact the predicting power of Laplacian energy to the critical pressures of alkanes is almost nil.
The sum connectivity energy shows similar correlation properties [ ]. The QSPR study in Table 10 reveals that the predicting power of sum connectivity energy is remarkably good. Infarct the sum connectivity energy can be use as a tool to predict the Heats of vaporization of alkanes. The correlation coefficient value of sum connectivity energy with the Heats of vaporization of alkanes is 0.931. Further the range of correlation coefficient value is 0.025 to 0.931. In fact, the predicting power of sum connectivity energy with Critical pressures of alkanes is almost nil.
The QSPR study of Vertex randic energy does not pass the test.
From practical point of view, topological indices for which the absolute values of correlation coefficient are less than 0.8 can be characterized as useless. Thus the QSPR study of 9 topological indices with physical properties of 68 alkanes helps us to characterize useful topological indices with absolute value of correlation coefficient lies between 0.8 to 0.972.