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QSPR Analysis of certain Distance Based Topological Indices

Shailaja Shirakol
Shailaja Shirakol
, 
Manjula Kalyanshetti
Manjula Kalyanshetti
 and 
Sunilkumar M. Hosamani
Sunilkumar M. Hosamani
   | Sep 27, 2019
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Published Online: Sep 27, 2019
Page range: 371 - 386
Received: Mar 05, 2019
Accepted: Apr 27, 2019
DOI: https://doi.org/10.2478/AMNS.2019.2.00032
Keywords
© 2019 Shailaja Shirakol et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 Public License.
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.

A fundamental concept of chemistry is that the structural characteristics of a molecule are responsible for its properties. Topological indices are a convenient means of translating chemical constitution into numerical values which can be used for correlation with physical properties in quantitative structure-property/activity relationship (QSPR/QSAR) studies. The use of graph invariant (topological indices) in QSPR and QSAR studies has become of major interest in recent years. Topological indices have found application in various areas of chemistry, physics, mathematics, informatics, biology, etc [1, 7, 26], but their most important use to date is in the non-empirical Quantitative Structure- Property Relationships (QSPR) and Quantitative Structure -Activity Relationships (QSAR) [5, 14, 19, 21, 23, 24, 27].
Survey of Selected Distance and Degree-Distance Based Topological Indices

Wiener Index: The Wiener index is named after Harry Wiener, who introduced it in 1947; at the time, Wiener called it the "path number"[24]. It is the oldest topological index related to molecular branching. Based on its success, many other topological indices of chemical graphs [2,3,13], based on information in the distance matrix of the graph, have been developed subsequently to Wiener’s work. Which is defined as:

let G be any connected graph of order n and size m. Then Wiener index of G is denoted by W(G) and is defined as follows.

W(G)=12∑u∈V(G)dG(u,v) $$W\left( G \right)=\frac{1}{2}\sum\limits_{\mathsf{u}\in V\left( G \right)}{{{d}_{G}}}\left( u,v \right)$$

Terminal Wiener Index: The concept of terminal Wiener index was put forward by Petrović et el. [9] some-what later but independently, Szekely et al. [25] arrived at the same idea. If G has k-pendent vertices labeled by v1;v2 . . .vk , then its terminal distance matrix is the square matrix of order k whose (i, j)-th entry is d(vi,vj \G). Terminal distance matrices were used for modeling amino acid sequences of proteins and of the genetic code [12, 17, 18].

The terminal Wiener index TW(G) of a connected graph G is defined as the sum of the distances between all pairs of its pendent vertices.

Thus if VT = {v1;v2; . . . ,vk} is the set of all pendent vertices of G, then

TW(G)=∑{ u,v⊆VT(G) }d(u,v\G)=∑1≤i<j≤kd(u,v\G) $$TW\left( G \right)=\sum\limits_{\left\{ u,v\subseteq {{V}_{T}}\left( G \right) \right\}}{d\left( u,v\backslash G \right)}=\sum\limits_{1\le i<j\le k}{d\left( u,v\backslash G \right)}$$

Degree Distance Index: The degree distance was introduced by Dobrynin and Kochetova [5] as a weighted version of the Wiener index. The degree distance of G, denoted by DD(G), is defined as follows

DD(G)=∑{ u,v }⊆V(G)dG(u,v)[ degG(u)+degG(v) ]. $$DD\left( G \right)=\sum\limits_{\left\{ u,v \right\}\subseteq V\left( G \right)}{{{d}_{G}}}\left( u,v \right)\left[ de{{g}_{G}}\left( u \right)+de{{g}_{G}}\left( v \right) \right].$$

Gutman Index: The Gutman index was put forward in [10] as a multiplicative version of degree-distance index which is defined as follows.

GI(G)=∑{ u,v }⊆V(G)dG(u,v)[ degG(u)degG(v) ]. $$GI\left( G \right)=\sum\limits_{\left\{ u,v \right\}\subseteq V\left( G \right)}{{{d}_{G}}}\left( u,v \right)\left[ de{{g}_{G}}\left( u \right)de{{g}_{G}}\left( v \right) \right].$$

Ashwini Index: Motivated by the terminal Wiener index Hosamani [15] has introduced a novel topological index viz, Ashwini index of a molecular graph G. Which is based on the terminal distance between any pair of pendant vertices together with their neighborhood degrees.

A T = ∑ 1 ≤ i < j ≤ n d T v i , v j d e g T N u i + d e g T N v j . $$\mathscr A\left( T \right)=\sum\limits_{1\le i<j\le n}{{{d}_{T}}}\left( {{v}_{i}},{{v}_{j}} \right)\left[ de{{g}_{T}}\left( N\left( {{u}_{i}} \right) \right)+de{{g}_{T}}\left( N\left( {{v}_{j}} \right) \right) \right].$$

Where N(v) = {u ∈ V(G) : uv ∈ E(G)}.

SM- Index: Motivated by the Gutman index and Ashwini index of a molecular graph G, we define here a new topological invariant namely SM-index of a molecular graph G. Which is defined as follows:

SM(T)=∑1≤i<j≤ndT(vi,vj)[ degT(N(ui))degTN(vj) ]. $$SM\left( T \right)=\sum\limits_{1\le i<j\le n}{{{d}_{T}}}\left( {{v}_{i}},{{v}_{j}} \right)\left[ de{{g}_{T}}\left( N\left( {{u}_{i}} \right) \right)de{{g}_{T}}N\left( {{v}_{j}} \right) \right].$$

Where N(v) = {u ∈ V(G) : uv ∈ E(G)}.

Hyper Wiener Index: In 1993, Milan Randić [20] introduced a distance based quantity, he named it as hyper Wiener index and denoted by WW. His definition could be applied only to trees, and was in possible to use for cycle-containing graphs.

WW(G)=12∑{ u,v }⊆V(G)[ dG(u,v)+dG2(u,v) ] $$WW\left( G \right)=\frac{1}{2}\sum\limits_{\left\{ u,v \right\}\subseteq V\left( G \right)}{\left[ {{d}_{G}}\left( u,v \right)+d_{G}^{2}\left( u,v \right) \right]}$$

which could be applied to all connected graphs, since then the above formula is used the definition of the hyper Wiener index.
The Use of Selected Distance and Degree-Distance Based Topological Indices in QSPR Studies

We have used three distance based topological indices and four degree-distance based topological indices viz,Wiener Index (W(G)),Terminal Wiener Index (TW(G)),Hyper Wiener Index (WW(G)) [distance baesd TI’s] and Degree-distance Index, Gutman Index (GI(G)),Ashwini Index A G , $\left( \mathscr A\left( G \right) \right),$SNM-Index [degree-distance based TI’s] 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 70 alkanes from n-butanes to nonanes. Values for these property were taken from Dejan Plavsić et. al [16]. The above said Distance and Degree-Distance topological indices and the experimental values for the physical properties of 70 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 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 1141.20 175.68 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 W(G) TW{G) DD{G) G1{G) A $\mathscr {A}$(G) SNM{G) HW{G)
1 Butane 10 3 28 19 12 12 46
2 2-methyl propane 9 6 20 13 36 54 27
3 Pentane 20 8 60 44 16 16 146
4 2-methyl butane 18 8 52 36 42 54 90
5 2,2 dimethylpropane 16 8 44 28 96 192 52
6 Hexane 35 5 110 82 20 20 371
7 2-methylpentane 32 10 96 73 52 66 254
8 3-methyalpentane 31 10 94 69 46 52 217
9 2,2-methylbutane 26 15 82 57 102 168 142
10 2,3-dimethylbutane 29 16 84 61 96 144 161
11 Heptanes 56 6 182 146 24 24 812
12 2-methylhexane 52 12 166 130 62 78 604
13 3-methylhexane 50 12 161 122 55 62 506
15 3-ethylpentane 48 12 150 144 48 48 408
16 2,2-dimethylpentane 49 16 142 106 120 192 370
17 2,3 -dimethylpentane 46 15 154 109 108 144 352
18 2,4-dimethylpentane 48 16 150 114 120 180 426
19 3,3-dimethylpentane 44 14 136 98 104 144 296
20 Octane 84 7 280 231 28 28 1596
21 2-methylheptane 79 14 263 211 72 90 1261
22 3 -methylheptane 76 14 248 212 64 72 1072
23 4-methylheptane 75 14 224 193 64 72 1011
24 3-ethylhexane 72 14 232 190 56 56 822
25 2,2-dimethylhexane 71 21 228 179 138 216 845
26 2,3 -dimethylhexane 71 22 291 275 118 156 766
27 2,4-dimethylhexane 71 23 231 179 125 168 803
28 2,5 -dimethylhexane 74 24 240 191 144 216 962
29 3,3-dimethylhexane 67 21 212 163 113 150 649
30 3,4-dimethylhexane 68 22 216 168 108 131 668
31 3-ethyl-2-methylpentane 67 22 215 163 108 130 607
32 3-ethyl-3-methylpentane 64 21 239 129 102 120 514
33 2,2,3 -trimethylpentane 63 27 196 147 198 318 495
34 2,2,4-trimethylpentane 66 32 194 152 228 330 606
35 2,3,3-trimethylpentane 60 27 242 200 194 314 458
36 2,3,4-trimethylpentane 65 32 198 155 192 288 551
37 Nonane 120 8 422 365 32 32 2892
38 2-methyloctane 114 16 391 327 82 102 2388
39 3-methyloctane 110 16 310 268 73 82 2076
40 4-methyloctane 108 16 353 295 73 82 1920
41 3-ethylheptane 105 16 331 271 72 72 1604
42 4-ethylheptane 102 16 239 190 68 68 1452
43 2,2-dimethylheptane 104 24 343 378 156 240 1718
44 2,3-dimethylheptane 102 25 336 264 133 174 1548
45 2,4-dimethylheptane 102 26 342 268 148 190 1524
46 2,5-dimethylheptane 110 27 344 269 147 198 1646
47 2,6- dimethylheptane 108 28 344 280 168 252 1926
48 3,3- dimethylheptane 98 24 320 277 138 184 1340
49 3,4- dimethylheptane 98 25 320 392 122 147 1298
50 3,5- dimethylheptane 100 26 320 264 128 156 1396
51 4,4- dimethylheptane 96 24 314 248 136 184 1218
52 3-ethyl-2-methylhexane 96 25 315 248 122 146 1146
53 4-ethyl-2-methylhexane 98 26 303 249 126 148 1244
54 3 -ethyl- 3-methylhexane 92 24 295 228 114 130 992
55 2,2,4- trimethylhexane 94 36 301 238 237 378 1108
56 2,2,5- trimethylhexane 98 38 322 270 270 474 1328
57 2,3,3- trimethylhexane 90 32 289 256 210 318 936
58 2,3,4- trimethylhexane 92 36 295 238 199 273 992
59 2,3,5- trimethylhexane 96 38 317 251 228 342 1188
60 3,3,4- trimethylhexane 87 34 278 214 201 278 838
61 3,3-diethylpentane 88 24 134 96 96 96 796
62 2,2-dimethyl-3-ethylpentane 88 32 279 213 208 304 814
63 2,3-dimethyl-3-ethylpentane 86 34 266 197 186 250 740
64 2,4-dimethyl-3-ethylpentane 90 36 291 224 200 276 870
65 2,2,3,3-tetramethylpentane 82 44 223 170 316 560 628
66 2,2,3,4- tetramethylpentane 86 47 273 209 327 564 758
67 2,2,4,4- tetramethylpentane 86 40 280 216 316 530 850
68 2,3,3,4- tetramethylpentane 84 47 264 200 28 464 729
Regression Models

We have tested the following linear regression model

P=A+B(TI) $$P=A+B\left( TI \right)$$

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.

Wiener indexW(G):

bp=20.8432+[ W(G) ]1.2203 $$bp=20.8432+\left[ W\left( G \right) \right]1.2203$$

mv=113.868+[ W(G) ]0.6412 $$mv=113.868+\left[ W\left( G \right) \right]0.6412$$

mr=24.7432+[ W(G) ]0.1944 $$mr=24.7432+\left[ W\left( G \right) \right]0.1944$$

hv=23.710+[ W(G) ]0.1998 $$hv=23.710+\left[ W\left( G \right) \right]0.1998$$

ct=179.262+[ W(G) ]1.4606 $$ct=179.262+\left[ W\left( G \right) \right]1.4606$$

cp=34.1143−[ W(G) ]0.1025 $$cp=34.1143-\left[ W\left( G \right) \right]0.1025$$

st=16.126748+[ W(G) ]0.0634 $$st=16.126748+\left[ W\left( G \right) \right]0.0634$$

mp=−129.02+[ W(G) ]0.2909 $$mp=-129.02+\left[ W\left( G \right) \right]0.2909$$

Terminal wiener index TW(G):

bp=68.2472+[ TW(G) ]1.9072 $$bp=68.2472+\left[ TW\left( G \right) \right]1.9072$$

mv=139.62+[ TW(G) ]1.002 $$mv=139.62+\left[ TW\left( G \right) \right]1.002$$

mr=32.25+[ TW(G) ]0.3178 $$mr=32.25+\left[ TW\left( G \right) \right]0.3178$$

hv=37.616+[ TW(G) ]0.055 $$hv=37.616+\left[ TW\left( G \right) \right]0.055$$

ct=230.099+[ TW(G) ]2.5196 $$ct=230.099+\left[ TW\left( G \right) \right]2.5196$$

cp=29.5512−[ TW(G) ]0.1327 $$cp=29.5512-\left[ TW\left( G \right) \right]0.1327$$

st=19.1536+[ TW(G) ]0.088 $$st=19.1536+\left[ TW\left( G \right) \right]0.088$$

mp=579.080−[ TW(G) ]32.24 $$mp=579.080-\left[ TW\left( G \right) \right]32.24$$

Hyper wiener index HW(G):

bp=69.8172+[ HW(G) ]0.04586 $$bp=69.8172+\left[ HW\left( G \right) \right]0.04586$$

mv=140.3322+[ HW(G) ]0.02365 $$mv=140.3322+\left[ HW\left( G \right) \right]0.02365$$

mr=33.2598+[ HW(G) ]0.00665 $$mr=33.2598+\left[ HW\left( G \right) \right]0.00665$$

hv=31.82212+[ HW(G) ]0.007513 $$hv=31.82212+\left[ HW\left( G \right) \right]0.007513$$

ct=245.468+[ HW(G) ]0.04589 $$ct=245.468+\left[ HW\left( G \right) \right]0.04589$$

cp=30.4719−[ HW(G) ]0.00433 $$cp=30.4719-\left[ HW\left( G \right) \right]0.00433$$

st=19.1519+[ HW(G) ]0.001937 $$st=19.1519+\left[ HW\left( G \right) \right]0.001937$$

mp=−120.603+[ HW(G) ]0.013596 $$mp=-120.603+\left[ HW\left( G \right) \right]0.013596$$

Degree distance index DD(G):

bp=32.2663+[ DD(G) ]0.3432 $$bp=32.2663+\left[ DD\left( G \right) \right]0.3432$$

mv=119.529+[ DD(G) ]0.182 $$mv=119.529+\left[ DD\left( G \right) \right]0.182$$

mr=37.18+[ DD(G) ]0.00714 $$mr=37.18+\left[ DD\left( G \right) \right]0.00714$$

hv=37.144+[ DD(G) ]0.00732 $$hv=37.144+\left[ DD\left( G \right) \right]0.00732$$

ct=196.428+[ DD(G) ]0.3932 $$ct=196.428+\left[ DD\left( G \right) \right]0.3932$$

cp=33.418−[ DD(G) ]0.0298 $$cp=33.418-\left[ DD\left( G \right) \right]0.0298$$

st=17.121+[ DD(G) ]0.0162 $$st=17.121+\left[ DD\left( G \right) \right]0.0162$$

mp=−122.787+[ DD(G) ]0.0641 $$mp=-122.787+\left[ DD\left( G \right) \right]0.0641$$

Gutman index GI(G):

bp=38.62+[ GI(G) ]0.3983 $$bp=38.62+\left[ GI\left( G \right) \right]0.3983$$

mv=201.965−[ GI(G) ]0.20958 $$mv=201.965-\left[ GI\left( G \right) \right]0.20958$$

mr=28.02+[ GI(G) ]0.061 $$mr=28.02+\left[ GI\left( G \right) \right]0.061$$

hv=26.978+[ GI(G) ]0.0632 $$hv=26.978+\left[ GI\left( G \right) \right]0.0632$$

ct=203.4429+[ GI(G) ].4577 $$ct=203.4429+\left[ GI\left( G \right) \right].4577$$

cp=20.218−[ GI(G) ]0.034733 $$cp=20.218-\left[ GI\left( G \right) \right]0.034733$$

st=17.37+[ GI(G) ]0.0190 $$st=17.37+\left[ GI\left( G \right) \right]0.0190$$

mp=−119.40+[ GI(G) ]0.0616 $$mp=-119.40+\left[ GI\left( G \right) \right]0.0616$$

Ashwini index A(G):

b p = 135.6962 + A G 0.1870 $$bp=135.6962+\left[ \mathscr A\left( G \right) \right]0.1870$$

m v = 148.45 + A G 0.1095 $$mv=148.45+\left[ \mathscr A\left( G \right) \right]0.1095$$

m r = 35.573 + A G 0.034 $$mr=35.573+\left[ \mathscr A\left( G \right) \right]0.034$$

h v = 39.95 + A G 0.00828 $$hv=39.95+\left[ \mathscr A\left( G \right) \right]0.00828$$

c t = 237.8298 + A G 0.3766 $$ct=237.8298+\left[ \mathscr A\left( G \right) \right]0.3766$$

c p = 28.4180 − A G 0.01427 $$cp=28.4180-\left[ \mathscr A\left( G \right) \right]0.01427$$

s t = 19.970 + A G 0.0078 $$st=19.970+\left[ \mathscr A\left( G \right) \right]0.0078$$

m p = − 113.0059 + A G 0.03 $$mp=-113.0059+\left[ \mathscr A\left( G \right) \right]0.03$$

SM index SM(G):

bp=97.968+[ SNM(G) ]0.069 $$bp=97.968+\left[ SNM\left( G \right) \right]0.069$$

mv=74.45+[ SNM(G) ]0.46 $$mv=74.45+\left[ SNM\left( G \right) \right]0.46$$

mr=10.394+[ SNM(G) ]0.152 $$mr=10.394+\left[ SNM\left( G \right) \right]0.152$$

hv=37.749+[ SNM(G) ]0.00587 $$hv=37.749+\left[ SNM\left( G \right) \right]0.00587$$

ct=268.506+[ SNM(G) ]0.0956 $$ct=268.506+\left[ SNM\left( G \right) \right]0.0956$$

cp=27.68−[ SNM(G) ]0.00582 $$cp=27.68-\left[ SNM\left( G \right) \right]0.00582$$

st=20.486+[ SNM(G) ]0.00266 $$st=20.486+\left[ SNM\left( G \right) \right]0.00266$$

mp=−109.163−[ SNM(G) ]0.00016 $$mp=-109.163-\left[ SNM\left( G \right) \right]0.00016$$

Discussion and Concluding Remarks

By inspection of the data given in tables 3 to 9, It is possible to draw numbers of conclusion for the given distance and degree-distance based TIs.

Statical parameters for the linear QSPR model for Wiener index.

Physical Properties N a b r s F
Boiling point 70 20.8432 1.2203 0.921 14.3278 388.436
Molar volume 67 113.868 0.6412 0.970 4.32084 1037.804
Molar refraction 67 24.7432 0.1944 0.962 1.45880 795.781
Heats of vaporization 67 23.710 0.1998 0.964 1.45328 846.841
Critical temperature 70 179.262 1.4606 0.899 19.8772 285.433
Critical Pressure 70 34.1143 -0.1025 0.921 1.2167 380.698
Surface tension 66 16.126748 0.0634 0.815 1.14383 126.812
Melting point 52 -129.02 0.2909 0.317 25.87537 5.585

Statical parameters for the linear QSPR model for terminal Wiener index.

Physical Properties N a b r s F
Boiling point 70 68.2472 1.9072 0.574 30.3815 33.333
Molar volume 67 139.62 1.002 0.606 14.17194 37.698
Molar refraction 67 32.25 0.3178 0.644 4.06101 46.075
Heats of vaporization 67 37.616 0.055 0.443 4.87876 15.910
Critical temperature 70 230.099 2.5196 0.620 35.5665 42.39
Critical Pressure 70 29.5512 -0.1327 0.475 2.7503 19.816
Surface tension 66 19.1536 0.088 0.481 1.73103 19.315
Melting point 52 579.080 -32.214 0.096 27.1532 0.461

Statical parameters for the linear QSPR model for hyper Wiener index.

Physical Properties N a b r s F
Boiling point 70 69.8172 0.04586 0.722 25.6758 73.879
Molar volume 67 140.3322 0.02365 0.759 11.59154 88.511
Molar refraction 67 33.2598 0.001937 0.555 1.64346 28.430
Heats of vaporization 67 31.82212 0.007513 0.789 3.34131 107.499
Critical temperature 70 245.468 0.04589 0.809 1.1358 129.100
Critical Pressure 70 30.4719 -0.00433 0.636 34.9872 6.078
Surface tension 66 19.1519 0.0019 0.555 1.64346 28.430
Melting point 52 -120.603 0.013596 0.318 25.86514 5.629

Statical parameters for the linear QSPR model for Degree distance index.

Physical Properties N a b r s F
Boiling point 70 32.2663 0.3432 0.863 18.7367 198.428
Molar volume 67 119.529 0.182 0.919 7.03409 315.874
Molar refraction 67 37.81 0.00714 0.897 2.34677 267.614
Heats of vaporization 67 37.144 0.00732 0.897 2.40325 268.443
Critical temperature 70 196.428 0.3932 0.809 26.6334 128.863
Critical Pressure 70 33.418 -0.0298 0.890 1.4243 259.425
Surface tension 66 17.121 0.0162 0.716 1.37956 67.174
Melting point 52 -122.787 0.0641 0.232 26.53927 2.839

Statical parameters for the linear QSPR model for Gutman index.

Physical Properties N a b r s F
Boiling point 70 38.62 0.3983 0.858 19.0533 189.648
Molar volume 67 201.965 -0.20958 0.892 8.05741 252.709
Molar refraction 67 28.02 0.061 0.874 2.58307 209.543
Heats of vaporization 67 26.978 0.0632 0.882 2.56979 226.624
Critical temperature 70 203.4429 0.4577 0.807 26.7911 126.553
Critical Pressure 70 20.218 -0.034733 0.889 1.4332 255.359
Surface tension 66 17.37 0.0190 0.711 1.38853 65.486
Melting point 52 -119.40 0.0616 0.187 26.80008 1.815

Statical parameters for the linear QSPR model for Ashwini index.

Physical Properties N a b r s F
Boiling point 70 135.6962 -0.1870 0.420 33.644 14.578
Molar volume 67 148.45 0.1095 0.492 15.51334 20.706
Molar refraction 67 35.573 0.034 0.519 4.53808 23.949
Heats of vaporization 67 39.95 -0.00823 0.288 5.21235 5.885
Critical temperature 70 237.8298 0.3766 0.459 40.2596 18.155
Critical Pressure 70 23.4180 -0.01427 0.381 2.8901 11.523
Surface tension 66 19.970 0.0078 0.320 1.87121 7.299
Melting point 52 -113.0059 0.03 0.561 4.90731 22.913

Statical parameters for the linear QSPR model for SM index.

Physical Properties N a b r s F
Boiling point 70 97.968 0.069 0.292 35.4737 6.3828
Molar volume 67 74.47 0.46 0.378 16.4949 10.809
Molar refraction 67 10.394 0.152 0.400 4.8866 12.352
Heats of vaporization 67 37.749 0.00587 0.158 5.37507 1.658
Critical temperature 70 268.506 0.0956 0.326 42.8387 8.093
Critical Pressure 70 27.68 -0.00582 0.288 2.9931 6.145
Surface tension 66 20.486 0.00266 0.198 1.93590 2.6614
Melting point 52 -109.163 -0.00016 0.339 5.1208 8.7442

First, the famous and much studied distance based Topological index viz, Wiener index found to be more suitable tool to predict the physical properties of alkanes.The Wiener Index shows good correlation with almost all physical properties of alkanes which are listed in table 3 except molar volume and surface tension of alkanes.The correlation coefficient value lies between 0.815 to 0.970. The QSPR study reveals that Wiener Index is more suitable to predict heats of vapourization and molar volumes of alkanes with correlation coefficient value r=0.964, and r = 0.0.970 respectively.

In addition the result for Terminal Wiener index revealed that the recent advocated idea of using Terminal Wiener index did not pass the test. This important details seems to have ignored in recent paper [9], on Terminal Wiener index.

Recently introduced distance based topological invariant viz, Hyper Wiener index found to be adequate for any structure-property correlation, except for critical temperatures of alkanes with correlation coefficient value r=0.809.

The QSPR study of degree-distance index in tables 6 reveals that the degree-distance index is an useful topological invariant. It shows good correlation with almost all physical properties which are listed in Table 6, except surface tension and melting points of alkanes. The correlation coefficient values lies between 0. 809 to 0.919. The degree-distance index is more suitable to predict the molar volume and heats of vaporization with r = 0 :919 and r = 0 :897 respectively.

The multiplicative version of degree-distance index is known as Gutman Index. By observing the results in table 7,One can say that the Gutman index has less predictive ability compared to degree-distance index. Further the correlation of Gutman index with physical properties of alkanes is very less and correlation coefficient value lies between 0.187 to 0.892.

The another degree-distance based topological index viz, Ashwini index. The predicting power of Ashwini index with physical properties of alkanes is too less. The correlation coefficient value of Ashwini index lies between 0.288 to 0.519.

Motivated by Gutman index and Ashwini index, Here we introduce a new degree-distance based topological invariant viz, SM Index. The QSPR study of SNM Index in table-9 shows good predicting power for alkanes.

From practical point of view, topological indices for which the absolute value of the correlation coefficient is less than 0.8 can be characterized useless. Thus the QSPR study of these distance and degree-distance based topological indices with physical properties of alkanes helps us to characterize useful topological indices indices with absolute values of correlation coefficients lies between 0.8 to 0.970.
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