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Chemical applicability of Gourava and hyper-Gourava indices
B. Basavanagoud, Shruti Policepatil Department of Mathematics, Karnatak University, Dharwad - 580 003, Karnataka, India [email protected], [email protected]
DOI 10.17586/2220-8054-2021-12-2-142-150
Topological indices are extensively used as molecular descriptors in building Quantitative Structure-Activity Relationship (QSAR), Quantitative Structure-Property Relationship (QSPR) and Quantitative Structure-Toxicity Relationship (QSTR). In this paper, Gourava and hyper-Gourava indices are tested with physico-chemical properties of octane isomers such as entropy, acentric factor and DHVAP using linear regression models. The first Gourava index highly correlates with entropy (coefficient of correlation 0.9644924) and the second Gourava index highly correlates with acentric factor (coefficient of correlation 0.962243). Further, Gourava and hyper-Gourava indices are obtained for the line graph of subdivision graph of 2D-lattice, nanotube and nanotorus of TUC4C8[p, q].
Keywords: topological indices, Gourava indices, hyper-Gourava indices, 2D-lattice of TUC4C8[p, q], TU C4C8[p, q] nanotube, TU C4C8 [p, q] nanotorus.
Received: 12 March 2021 Revised: 17 March 2021
1. Introduction
Let G = (V, E) be a graph with a vertex set V(G) and an edge set E(G) such that | V(G) | = n and |E(G) | = m. The degree of a vertex dG(v) is the number of edges incident to it in G. For undefined graph theoretic terminologies and notations refer to [1] or [2]. The line graph [1] L(G) of a graph G with vertex set as the edge set of G and two vertices of L(G) are adjacent whenever the corresponding edges in G have a vertex incident in common. The subdivision graph [1] S(G) of a graph G whose vertex set is V(G) U E(G) where two vertices are adjacent if and only if one is a vertex of G and other is an edge of G incident with it.
Topological indices promise to have far-reaching applications in drug design, cancer research and bonding theory etc. Among them, first degree based topological index is first Zagreb index developed in 1972 [3]. Further, the second Zagreb index [4] and F-index [5] were studied. Motivated by the definitions of the Zagreb indices and their wide applications, Kulli introduced the first Gourava index and second Gourava index of a molecular graph [6] as follows:
GOi (G) = ^ [dG(u) + dG(v) + dG(u)dG(v)},
uveE(G)
GO2(G)= ^ [(dG(u)+ dG(v))(dG(u)dG(v))].
uveE(G)
In [7], Kulli introduced the first and second hyper-Gourava indices of a molecular graph G which are defined as:
HGOi(G)= ^ [dG (u) + dG(v) + dG(u)dG(v)]2,
uveE(G)
HGO2(G)= ^ [(dG(u) + dG(v))(dG(u)dG(v))]2.
uveE(G)
Alkanes are the important organic compounds classified under hydrocarbons and they are saturated, that is the carbon backbone consists of carbon to carbon single bonds only. There are a wide variety of alkanes that play vital roles in our daily life. Alkanes are used in LPG (Liquefied Petroleum Gas), propellants, disposable lighters etc. Alkanes containing 5-8 carbon atoms are used as fuel and as good solvents for nonpolar substances. For this purpose, they are widely used in industries and research work. Two or more compounds having same molecular formula but different chemical structures are called isomers. Eighteen isomers of octane are depicted in Fig. 1.
A nanostructure is an object that has at least one dimension equal to or smaller than 100 nanometers. There are various types of nanostructures such as nanoparticles, nanotubes and nanopores etc., In the last few years, much research has been concentrated on the use of nanostructures towards problems of biology and medicine. Topological index of TURC4C8(S) nanotube, armchair polyhex nanotube, V-phenylenic nanotube and V-phenylenic nanotori are discussed in [8]. For more discussion on nanostructures, readers are referred to [9-11].
(a)
(m)
(n)
(o)
(P)
(?)
(r)
Fig. 1. (a) n-octane; (b) 2-methyl heptane; (c) 3-methyl heptane; (d) 4-methyl heptane; (e) 3-ethyl hexane; (f) 2,2-dimethyl hexane; (g) 2,3-dimethyl hexane; (h) 2,4-dimethyl hexane; (i) 2,5-dimethyl hexane; (j) 3,3-dimethyl hexane; (k) 3,4-dimethyl hexane; (l) 3-ethyl-2-methyl pentane; (m) 3-ethyl-3-methyl pentane; (n) 2,2,3-trimethyl pentane; (o) 2,2,4-trimethyl pentane; (p) 2,3,3-trimethyl pentane; (q) 2,3,4-trimethyl pentane; (r) 2,2,3,3-tetramethyl butane.
2. Chemical applicability of Gourava and hyper-Gourava indices
In this section, we present a linear regression model of these physical properties with the Gourava and hyper-Gourava indices. Gourava and hyper-Gourava indices are degree based indices. These indices have good correlation with physical properties of chemical compounds like entropy (S), acentric factor (AcentFac) and standard enthalpy of vaporization (DHVAP) of octane isomers. Gourava and hyper-Gourava indices are tested using a data set of octane isomers found at http://www.moleculardescriptors.eu/dataset.htm. The columns 5, 6, 7 and 8 of Table 2 are computed by using the definition of first Gourava index, second Gourava index, first hyper-Gourava index and second hyper-Gourava index, respectively.
The linear regression models for entropy, acentic factor and DHVAP using the data of Tables 2 are obtained using the least squares fitting procedure as implemented in R software [12].
Gourava indices values against entropy, acentric factor and DHVAP values are plotted in Figs. 2,4 and 6. Hyper-Gourava indices values against entropy, acentric factor and DHVAP values are plotted in Figs. 3, 5 and 7.
Fig. 2. Scatter diagram of S on GO1 (left) and GO2(right), superimposed by the fitted regression line The fitted models for GO1 are:
S = 137.6(±2.2) - 0.52(±0.04)GO1, (2.1)
AcentFac = 0.6(±0.02) - 0.004(±0.0003)GO1? (2.2)
DHVAP = 11.5(±0.4) - 0.04(±0.006)GO1. (2.3) The fitted models for GO2 are:
S = 120.5(±1.2) - 0.1(±0.008)GO2, (2.4)
AcentFac = 0.5(±0.009) - 0.0008(±0.00006)GO2, (2.5)
DHVAP = 10.2(±0.2) - 0.007(±0.001)GO2. (2.6)
Table 1. Experimental values of entropy, acentric factor, DHVAP and the corresponding values of first Gourava index, second Gourava index, first hyper-Gourava index and second hyper-Gourava index of octane isomers
Alkane S AcentFac DHVAP GO1 GO2 HGO1 hgo2
n-octane 111.67 0.397898 9.915 50 92 370 1352
2-methyl-heptane 109.84 0.377916 9.484 54 108 436 1992
3-methyl-heptane 111.26 0.371002 9.521 55 116 469 2528
4-methyl-heptane 109.32 0.371504 9.483 55 116 469 2528
3-ethyl-hexane 109.43 0.362472 9.476 56 124 502 3064
2,2-dimethyl-hexane 103.42 0.339426 8.915 62 146 592 4052
2,3-dimethyl-hexane 108.02 0.348247 9.272 60 142 582 4540
2,4-dimethyl-hexane 106.98 0.344223 9.029 59 132 535 3168
2,5-dimethyl-hexane 105.72 0.35683 9.051 58 124 502 2632
3,3-dimethyl-hexane 104.74 0.322596 8.973 64 164 668 5736
3,4-dimethyl-hexane 106.59 0.340345 9.316 61 150 615 5076
2-methyl-3-ethyl-pentane 106.06 0.332433 9.209 61 150 615 5076
3-methyl-3-ethyl-pentane 101.48 0.306899 9.081 66 182 744 7420
2,2,3-trimethyl-pentane 101.31 0.300816 8.826 69 192 799 9336
2,2,4-trimethyl-pentane 104.09 0.30537 8.402 66 162 658 4692
2,3,3-trimethyl-pentane 102.06 0.293177 8.897 70 202 842 3428
2,3,4-trimethyl-pentane 102.39 0.317422 9.014 65 168 695 6552
2, 2, 3,3-tetramethyl-butane 93.06 0.255294 8.41 88 282 1202 20724
400 600 300 1000 1200 5000 10000 1 5000 20000
HGOi HGOj
Fig. 3. Scatter diagram of S on HGOi (left) and HGO2 (right), superimposed by the fitted regression line
Fig. 4. Scatter diagram of Acent Fac on GO1(left) and GO2(right), superimposed by the fitted regression line
Fig. 5. Scatter diagram of Acent Fac on HGO1(left) and HGO2(right), superimposed by the fitted regression line
Fig. 6. Scatter diagram of DHVAP on GO1(left) and GO2(right), superimposed by the fitted regression line
400 600 300 1000 1200 5000 10000 1 5000 20000
HGOl
Fig. 7. Scatter diagram of DHVAP on HGO1(left) and HGO2(right), superimposed by the fitted regression line
The fitted models for HGO1 are:
S = 119.5(±1.2) - 0.02(±0.002)HGO1, (2.7)
AcentFac = 0.4(±0.009) - (0.0002)(±0.00001)HGO1, (2.8)
DHVAP = 10.1(±0.2) - 0.002(±0.0003)HGO;i^. (2.9) The fitted models for HGO2 are:
S = 110.1(±0.9) - 0.0009(±0.0001)HGO2, (2.10)
AcentFac = 0.4(±0.01) - 0.00001(±0.000001)HGO2, (2.11)
DHVAP = 9.4(±0.1) - 0.00006(±0.00002)HGO2. (2.12)
Note: The values in brackets of Eqns. (2.1) to (2.12) are the corresponding standard errors of the regression coefficients. The index is better as |r| approaches 1.
From Table 2, Figs. 2, 4 and 6 we can observe that GOi correlates highly with entropy and the correlation coefficient |r| = 0.9644924. Also, GO1 has good correlation (|r| > 0.9) with Acentric Factor and (|r| > 0.8) with DHVAP.
From Table 3, Figs. 2,4 and 6 we can observe that GO2 correlates highly with Acentric Factor and the correlation coefficient |r| = 0.9644924. Also, GO2 has good correlation (|r| > 0.9) with entropy and (|r| > 0.75) with DHVAP.
From Table 4, Figs. 3, 5 and 7 we can observe that HGO1 correlates highly with Acentric Factor and the correlation coefficient |r| = 0.9554303. Also, HGO1 has good correlation (|r| > 0.9) with entropy and (|r| > 0.75) with DHVAP.
From Table 5, Figs. 3, 5 and 7 HGO2 has good correlation (|r| > 0.85) with entropy, (|r| > 0.75) with Acentric Factor and (|r| > 0.6) with DHVAP.
TABLE 2. Correlation coefficient and residual standard error of regression models for GO1
Physical properties Absolute value of the correlation coefficient (|r|) Residual standard error
Entropy 0.9644924 1.23
Acentric Factor 0.9595891 0.01028
DHVAP 0.8368024 0.2163
Table 3. Correlation coefficient and residual standard error of regression models for GO2
Physical properties Absolute value of the correlation coefficient (|r|) Residual standard error
Entropy 0.9561406 1.364
Acentric Factor 0.962243 0.009946
DHVAP 0.7896431 0.2424
Table 4. Correlation coefficient and residual standard error of regression models for HGOi
Physical properties Absolute value of the correlation coefficient (|r|) Residual standard error
Entropy 0.953055 1.41
Acentric Factor 0.9554303 0.01079
DHVAP 0.779148 0.2477
Table 5. Correlation coefficient and residual standard error of regression models for HGO2
Physical properties Absolute value of the correlation coefficient (|r|) Residual standard error
Entropy 0.8691773 2.303
Acentric Factor 0.798397 0.022
DHVAP 0.6442586 0.3022
3. Gourava and hyper-Gourava indices of line graph of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8 [p, q]
In this section, we obtain the Gourava and hyper-Gourava indices of line graph of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8 [p, q]. Let p and q denote the number of squares in a row and the number of rows of squares, respectively in 2D-lattice, nanotube and nanotorus of TUC4C8[4, 3] as shown in Fig 8 (a), (b) and (c) respectively.
(a) (b) (c)
FIG. 8. (a) 2D-lattice of TUC4C [4,3]; (b) TUC^C^, 3] nanotube; (c) TUCiC^, 3] nanotorus The number of vertices and edges of 2D-lattice, nanotube and nanotorus of TUC4C8 [p, q] are given in Table 6.
Table 6. Order and size of graphs
Graph Order Size
2D-lattice of TUC4C [p, q] 4pq 6pq — p — q
TUC4C8[p, q] Nanotube 4pq 6pq-p
TUC4C8[p, q] Nanotorus 4pq 6pq
For more details about topological indices of the line graph of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p, q], refer [9-11].
The following theorem gives the Gourava and hyper-Gourava indices of line graph of subdivision graphs of 2D-latticeof TUCiC8[p, q].
Theorem 3.1. Let A be the line graph of the subdivision graph of2D-lattice of TU C4C8 [p, q] (See Fig. 9). Then
1. GO1(A) = 270pq - 105(p + q) + 4,
2. GO2(a) = 972pq - 442(p + q) + 40,
3. HGO1(A) = 4050pq - 1863(p + q) + 188,
4. HGO2(a) = 52488pq - 27964(p + q) + 5488.
Proof. Let A be the line graph of the subdivision graph of 2D-lattice of TUC4 C8[p, q]. The graph A has 2(6pq - 6-q) vertices and 18pq - 5p - 5q edges.
1. By using definition of first Gourava index and information in Table 7, we have:
GOi(A) = ^ [(dA(u) + dA(v)) + (dA(u)dA (v))]
(u,v)eE(A)
= |E2,2|[(2 + 2) + (2 x 2)] + |E2,3|[(2 + 3) + (2 x 3)] + |Ез,з|[(3 + 3) + (3 x 3)] = 2(p + q + 2)(4 + 4) + 4(p + q - 2)(5 + 6) + (18pq - 11p - 11q + 4)(6 + 9) = 270pq - 105(p + q) + 4.
2. By using definition of second Gourava index and information in Table 7, we have:
GO2(A) = ^ [(dA(u) + dA(v))(dA(u)dA(v))]
(u,v)eE(A)
= |E2,2|[(2 + 2)(2 x 2)] + |Е2,з|[(2 + 3)(2 x 3)] + |Ез,з|[(3 + 3)(3 x 3)] = 2(p + q + 2)(4 x 4) + 4(p + q - 2)(5 x 6) + (18pq - 11p - 11q + 4)(6 x 9) = 972pq - 442(p + q) + 40.
3. By using definition of first hyper-Gourava index and information in Table 7, we have:
HGOi (A) = ^ [(dA(u) + dA(v)) + (dA(u)dA (v))]2
(u,v)eE(A)
= |E2,2|[(2 + 2) + (2 x 2)]2 + |Е2,з|[(2 + 3) + (2 x 3)]2 + |Ез,з|[(3 + 3) + (3 x 3)]2 = 2(p + q + 2)(4 + 4)2 + 4(p + q - 2)(5 + 6)2 + (18pq - 11p - 11q + 4)(6 + 9)2 = 4050pq - 1863(p + q) + 188.
4. By using definition of second hyper-Gourava index and information in Table 7, we have:
HGO2(A) = ^ [(dA(u) + dA(v))(dA(u)dA(v))]2
(u,v)eE(A)
= |E2,2|[(2 + 2)(2 x 2)]2 + |Е2,з|[(2 + 3)(2 x 3)]2 + |Ез,з|[(3 + 3)(3 x 3)]2 = 2(p + q + 2)(4 x 4)2 + 4(p + q - 2)(5 x 6)2 + (18pq - 11p - 11q + 4)(6 x 9)2 = 52488pq - 27964(p + q) + 5488.
□
(a) (b)
Fig. 9. (a) Subdivision graph of 2D-lattice of TUC4C8[4,3]; (b) Line graph of the subdivision graph of 2D-lattice of TUC4C8 [4, 3]
The following theorem gives the Gourava and hyper-Gourava indices of line graph of subdivision graphs of TUG4G8[p, q] of nanotube.
Table 7. The edge partition of A based on degree of each edge
dA(u),dA(v) : uv G E(A) (2, 2) (2, 3) (3,3)
Number of edges 2(p + q + 2) 4(p + q - 2) 18pq - 11p - 11q + 4
Theorem 3.2. Let B be the line graph of the subdivision graph of TUC4C8 [p, q] nanotube (See Fig. 10). Then
1. GOi(B) = 270pq - 105p,
2. GO2(b) = 972pq - 442p,
3. HGO1(B) = 4050pq - 1863p,
4. HGO2(b) = 52488pq - 27964p.
Proof. By using the definition of Gourava and hyper-Gourava indices and information given in Table 8, we obtain the desired result. □
(a) (b)
Fig. 10. (a) Subdivision graph of TUC4C8[4, 3] of nanotube; (b) Line graph of the subdivision graph of TUC4C8 [4,3] of nanotube
Table 8. The edge partition of B based on degree of each edge
db (u), ds (v) : uv G E(B) (2, 2) (2, 3) (3, 3)
Number of edges 2p 4p 18pq - 11p
The following theorem gives the Gourava and hyper-Gourava indices of line graph of subdivision graphs of
TUC4C8[p, q] of nanotorus.
Theorem 3.3. Let C be the line graph of the subdivision graph of TUC4 C8 [p, q] nanotorus (See Fig. 11). Then
1. GO1(C) = 270pq,
2. GO2(c) = 972pq,
3. HGO1(C) = 4050pq,
4. HGO2(c) = 52488pq.
Proof. By using the definition of Gourava and hyper-Gourava indices and information given in Table 9, we obtain the desired result. □
Table 9. The edge partition of C based on degree of each edge
dc(u),dc(v) : uv G E(C) (3, 3)
Number of edges 18pq
(a) (b)
Fig. 11. (a) Subdivision graph of TUC4C8[4,3] of nanotorus; (b) Line graph of the subdivision graph of TUC4C8 [4,3] of nanotorus
4. Conclusion
In this paper, we have studied the chemical applicability of Gourava and hyper-Gourava indices. These indices show good correlation with physico-chemical properties. The first Gourava index highly correlates with entropy (coefficient of correlation is 0.9644924 and residual standard error is 1.23) which is better than the modified first Kulli-Basava index (|r| = 0.9476403 and residual standard error is 1.363) [13], the first Kulli-Basava index (|r| = 0.956207 and residual standard error is 1.415) [13] and the first neighbourhood Zagreb index (|r| = 0.9526144 and residual standard error is 1.416) [14]. The second Gourava index highly correlates with acentric factor (coefficient of correlation is 0.962243 and residual standard error is 0.009946) which is better than (3, a) connectivity index (|r| = 0.95802 and residual standard error is 0.01047) [15]. In addition, we computed Gourava and hyper-Gourava indices of the line graph of subdivision graph of 2D-lattice, nanotube and nanotorus of TUC4C8 [p, q]. These results would help to understand the chemical reactivity and biological activity of the nanostructures.
Acknowledgements
B. Basavanagoud is supported by the University Grants Commission (UGC), New Delhi, through UGC-SAP DRS-III for 2016-2021: F.510/3/DRS-III/2016(SAP-I).
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