View of The Pivotal Role of Degree-Based and Neighborhood Degree-Sum-Based Topological Indices in Predicting the Physico-Chemical Properties of N-Octane Isomers

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http://annalsofrscb.ro 4481

The Pivotal Role of Degree-Based and Neighborhood Degree-Sum-Based Topological Indices in Predicting the Physico-Chemical Properties of

N-Octane Isomers

Tamilarasi.C

Hindustan Institute of Technology and Science, Chennai, Tamilnadu, India.

Simon Raj.F

Hindustan Institute of Technology and Science, Chennai, Tamilnadu, India.

ABSTRACT

Neighborhood degree-sum topological index is sum of the degree of neighborhood vertices of the vertex u and is denoted as S_u where degree-based topological index is the number of edges meet at u and is denoted as d_u. In this paper, we have done the comparative analysis between ten notable degree-based indices with their corresponding neighborhood degree-sum topological indices. We report here, the correlation of mentioned types of topological indices with five Physico-chemical properties of octane isomers. Furthermore, we have discussed a deeper-lying relation between Augmented Zagreb Index with Atom-bond Connectivity Index based on the correlation with one of the Physico-chemical properties (boiling point) of octane isomers.

Keywords: Neighborhood degree-sum topological indices, degree of vertex, Physico-Chemical properties.

Introduction

Graph theory is the main branch of discrete mathematics which involves many applications to modeling real- life problems. Chemical graph theory has an important role in QSPR analysis through topological indices. Topological indices are the numbers associated with molecular graphs where atoms denote vertices and their respective chemical bonds denote edges.

Degree-based topological index is the main branch of topological indices. It includes neighborhood degree-sum topological indices whose applications have been widespread in recent years. Eighteen octane structural isomers have been used in the study of QSPR/QSAR due to their structural discrimination and the availability of experimental values (Physico- Chemical properties). In 1975, Milan Randic set forth the first degree-based topological index in his seminal paper “On Characterization of molecular branching”. Zagreb indices are one of the oldest degree-based topological indices in analyzing the structuredependence of total π- electron energy. Neighborhood degree is the summation of the degree of the nearest(neighborhood) vertices and is denoted as Su whereas the degree of a vertex is the number of edges meet at the vertex u and is denoted as du.

In chemical applicability, many degree-based as well as neighborhood degree-sum based topological indices are employed. It began in 1947 when the Wiener index was used to determine the physical properties of alkanes. One of the main applications of topological indices is to predict[19] the Physico-chemical properties of chemical compounds. Here we compare the predicting ability of the existing degree-based topological indices and their corresponding neighborhood degree-sum based topological indices like QSPR models to predict the properties[18] (boiling point- BP, entropy-S, acentric factor, enthalpy of vaporization-HVAP, standard enthalpy of vaporization-DHVAP) of octane isomers.

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http://annalsofrscb.ro 4482 Preliminaries

Let G be a simple connected graph with vertex set V(G) and edge set E(G). Throughout this paper, we have taken molecular graph G where the vertex set contains the atoms as vertices and edge set contains connectors(bonds) as edges.

Degree d_u(G)is considered as d_u(G) = {u𝜖𝑉(𝐺)/ 𝑢𝑣𝜖𝐸(𝐺)}

Neighborhood degree-sum Sᵤis considered as

Sᵤ(G) = _{𝑢𝜖𝑁 (𝐺)}𝑑ᵤ where N(G) is the set of neighborhood vertices of u.

Table 1 shows the basic definitions of degree-based & their corresponding neighborhood-degree-based topological indices.

No Degree-based topological indices Neighborhood degree-sum topological indices 1

[7] Augmented Zagreb Index Az(G) = ^{𝑑ᵤ 𝑑ᵥ}

𝑑ᵤ+𝑑ᵥ−2 3 𝑢,𝑣 ∊𝐸

[13] Sanskruti Index S(G)= ^{𝑆ᵤ 𝑆ᵥ}

𝑆ᵤ+𝑆ᵥ−2 3 𝑢,𝑣∊𝐸

2

[2] Atom Bond Connectivity Index ABC(G) = ^{𝑑ᵤ+𝑑ᵥ−2}

𝑑ᵤ 𝑑ᵥ 𝑢,𝑣𝜖𝐸

Fourth version of Atom Bond Connectivity Index ABC₄(G) = ^{𝑆ᵤ+𝑆ᵥ−2}

𝑆ᵤ 𝑆ᵥ 𝑢,𝑣𝜖𝐸

3

[16] Geometric Arithmetic Index GA(G) = ^{2 𝑑ᵤ 𝑑ᵥ}

𝑑ᵤ+𝑑ᵥ 𝑢,𝑣𝜖𝐸

Fifth version of Geometric Arithmetic Index GA5(G) = ^{2 𝑆ᵤ 𝑆ᵥ}

𝑆ᵤ+𝑆ᵥ 𝑢,𝑣𝜖𝐸

4 Harmonic Index H(G) = ²

𝑑ᵤ+𝑑ᵥ 𝑢,𝑣𝜖𝐸

[17] Neighborhood version of Harmonic Index NH(G) = ²

𝑆ᵤ+𝑆ᵥ 𝑢,𝑣𝜖𝐸

5

[12] Forgotten Index

F(G) = _{𝑢,𝑣𝜖𝐸} 𝑑ᵤ ²+ (𝑑ᵥ)² Or F(G) = _𝑢𝜖𝑉 𝑑ᵤ ³

[17] Neighborhood version of Forgotten Index NF*(G) = _{𝑢,𝑣𝜖𝐸} 𝑆ᵤ ²+ (𝑆ᵥ)² and NF(G) = _𝑢𝜖𝑉 𝑆ᵤ ³

6

[5] First Zagreb Index

M₁(G) = _{𝑢,𝑣𝜖𝐸}𝑑ᵤ + 𝑑ᵥ Or M₁(G) = _𝑢𝜖𝑉 𝑑ᵤ ³

[17] Neighborhood version of First Zagreb Index NM₁*(G) = _{𝑢,𝑣𝜖𝐸}𝑆ᵤ + 𝑆ᵥ and NM₁(G)₌ _𝑢𝜖𝑉 𝑆ᵤ ²

7 [5] Second Zagreb Index M₂(G) = _{𝑢,𝑣𝜖𝐸}𝑑ᵤ𝑑ᵥ

[17] Neighborhood version of Second Zagreb IndexNM₂(G) = 𝑢,𝑣𝜖𝐸𝑆ᵤ𝑆ᵥ

8 [3] Randic Index R(G) = ¹

𝑑ᵤ𝑑ᵥ 𝑢,𝑣 ∈𝐸

Neighborhood version of Randic Index NR(G) =

1 𝑆ᵤ𝑆ᵥ 𝑢,𝑣 ∈𝐸

9 [3] Sum Connectivity Index S(G) =

1 𝑑ᵤ+ 𝑑ᵥ 𝑢,𝑣 ∈𝐸

Neighborhood version of Sum Connectivity Index NS(G)

= ¹

𝑆ᵤ+ 𝑆ᵥ 𝑢,𝑣 ∈𝐸

10 [10] Hyper Zagreb IndexHM(G) = (𝑑ᵤ + 𝑑ᵥ)²

𝑢,𝑣𝜖𝐸

[17] Neighborhood version of Hyper Zagreb Index NHM(G) = 𝑢,𝑣𝜖𝐸 (𝑆ᵤ + 𝑆ᵥ)²

Table 2 and 3 show the results of the ten degree-based topological indices and their corresponding neighborhood degree-sum degree based topological indices. Forgotten Index

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http://annalsofrscb.ro 4483 and First Zagreb Index are calculated by vertex as well as edge partition methods. Their corresponding neighborhood degree-sum topological indices(NF*(G) ≠ 𝑁𝐹(𝐺) &NM₁*(G)

≠ 𝑁𝑀1(𝐺)) give unequal values. Table 4 shows the results of correlation between the basic Physico-chemical properties(BP,S, Acentric factor, HVAP & DHVAP) of n-octane isomers with mentioned degree-based as well as neighborhood degree-based topological indices.

Table 2. Shows the results of degree-based topological indices with n-octane isomers

Octane

isomers Az(G) ABC(G) GA(G) H(G) F(G) M1(G) M2(G) R(G) SUM(G )

HM(G )

56 4.94977 6.88562 3.83333 50 26 24 3.91422 3.6547 98

46.75 5.16855 5.67486 3.56667 62 28 26 3.77006 3.52456 114

51.375 5.05916 6.71124 3.63334 62 28 27 3.80807 3.54912 116

56 4.94977 6.76782 3.7 62 28 28 3.84608 3.57368 118

39.1111 5.42653 6.28562 3.2 92 32 30 3.56067 3.32723 152

45.51563 5.2375 6.52069 3.4 74 30 30 3.68074 3.43281 134

42.125 5.27794 6.43028 3.36667 74 30 29 3.66391 3.41898 132

37.5 5.38733 6.42369 3.3 74 30 28 3.6259 3.39442 130

44.74074 5.26761 6.37124 3.30006 92 32 32 3.62134 3.36562 156

50.14063 5.12811 6.57726 3.20482 74 30 31 3.71875 3.45732 136

50.37037 5.10869 6.45686 2.73333 92 32 34 3.68201 3.40401 160

40.31011 5.47431 6.17838 3.05238 104 34 35 3.48139 3.24415 174

29.86111 5.64531 6.05466 2.93333 104 34 32 3.41651 3.19709 168

41.31471 5.42478 6.70741 3.08571 104 34 36 3.50405 3.25798 176

39.65626 5.41584 6.33012 3.16667 86 32 33 3.55342 3.3165 152

33.18518 5.80855 5.8 2.65 134 38 40 3.25 3.03682 214

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http://annalsofrscb.ro 4484 Table 3. shows the results of neighborhood-degree-based topological indices with n-

octane isomers

Octane isomers

S ABC₄ GA₅ NH N*F NF N*

M1

N M1

N M2

NR NSUM N

HM 100.537 4.54233 6.93908 2.12143 172 326 48 90 84 2.14386 2.71099 340 115.0822 4.4392 6.93664 1.95158 202 406 52 104 98 1.97151 2.60130 398 122.61378 4.44109 6.86822 1.91348 224 448 54 108 106 1.9559 2.56987 436

123.91016 4.44298 6.86435 1.95000 228 472 54 110 107 1.9911 2.58753 442

124.14258 4.45621 6.78913 1.89849 252 520 56 114 115 1.96436 2.55123 482

155.83184 4.25251 6.92663 1.73420 274 632 60 138 132 1.75445 2.44290 538

145.952 4.32354 6.81908 1.73586 282 582 60 126 129 1.78369 2.44421 540

138.63364 4.34725 6.86001 1.75880 258 558 58 124 121 1.79941 2.46879 500

132.46078 4.33091 6.94658 1.78730 232 486 56 118 113 1.80194 2.49494 458

170.69576 4.2673 6.78158 1.65642 220 728 64 146 148 1.71766 2.38115 620

150.66144 4.34319 6.73084 1.67778 306 630 62 130 136 1.75341 2.40431 578

163.11248 4.31513 6.56525 1.63417 330 727 64 141 146 1.71322 2.37217 622

182.73748 4.29184 6.40313 1.53896 374 806 68 152 163 1.64476 2.30143 700

192.86378 4.13509 6.75786 1.46900 384 850 70 162 171 1.52741 2.25203 726

170.60628 4.1673 6.83317 1.58659 312 778 64 156 147 1.6062 2.34610 606

200.76881 4.1345 6.62052 1.41431 408 874 72 164 179 1.48892 2.21368 766

166.33789 4.21608 6.60986 1.53333 342 728 66 144 151 1.58908 2.30779 644

248.75879 3.89657 6.77254 1.23377 488 107 0

80 194 217 1.27674 2.07632 922

Comparative Analysis

(a) Between degree-based & neighborhood degree-based topological indices in correlation with Physico-chemical properties of n-octane isomers.

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http://annalsofrscb.ro 4485 (b) Between Augmented Zagreb Index with Atom-bond Connectivity Index based on

the correlation with boiling point of octane isomers.

(a) Based on the results shown in table 2 & 3, the ten degree-based topological indices and their corresponding twelve neighborhood degree-based topological indices are correlated with the basic Physico-chemical properties [19, 20]of n-octane isomers(Table 4).

Table 4. shows the results of correlation coefficients between the physicochemical properties of n-octane isomers with degree-based as well as neighborhood degree-sum

based topological indices.

Topological Index BP S Acentric Factor HVAP DHVAP

Az(G) 0.9223 0.6863 0.6676 0.9378 0.8997

S(G) -0.5352 -0.9556 -0.9844 -0.7504 -0.831

ABC(G) -0.8631 -0.8207 -0.7929 -0.9298 -0.9253

ABC₄(G) 0.6998 0.953 0.9594 0.8611 0.913

GA(G) 0.5964 0.5219 0.4425 0.6440 0.5911

GA₅(G) -0.1126 0.4389 0.5518 0.1529 0.2317

H(G) 0.6154 0.9124 0.9253 0.8121 0.8577

NH(G) 0.5696 0.9334 0.9857 0.7844 0.8520

F(G) -0.7047 -0.9528 -0.965 -0.8716 -0.924 NF*(G) -0.398 -0.8888 -0.9227 -0.6287 -0.7135 NF(G) -0.5493 -0.9300 -0.9912 -0.7636 -0.8412

M₁(G) -0.7203 -0.9543 -0.9731 -0.886 -0.9361

NM₁*(G) -0.4951 -0.9361 -0.9850 -0.7222 -0.8063

NM₁(G) -0.6235 -0.9475 -0.9947 -0.8181 -0.8875

M₂(G) -0.5007 -0.9410 -0.9864 -0.7281 -0.8118

NM₂(G) -0.4955 -0.9427 -0.9844 -0.7191 -0.8063

R(G) 0.8208 0.9061 0.9042 0.9361 0.9582

NR(G) 0.6326 0.9456 0.9839 0.8300 0.8881

SUM (G) 0.8023 0.9231 0.9299 0.9318 0.9612 NSUM(G) 0.5517 0.9364 0.9879 0.7699 0.8881 HM(G) -0.6567 -0.9613 -0.9829 -0.8425 -0.9043 NHM(G) -0.4591 -0.9319 -0.9787 -0.6917 -0.7818

From the following paragraphs, predictive ability[22,23] of the above mentioned ten topological indices and their corresponding twelve neighborhood degree-sum topological

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http://annalsofrscb.ro 4486 indices are tested with basic Physico-chemical properties of eighteen structural octane isomers are discussed.

Boiling Point(BP) Augmented Zagreb Index and Atom Bond Connectivity Index show higher correlation(0.9223 & 0.8631) than other degree based topological indices. Besides their corresponding neighborhood degree-based topological indices show moderated correlation withboiling point of n-octane isomers. The important analysis regarding the correlation, degree based topological indices are higher than their corresponding neighborhood-degree-based topological indices.

Entropy(S) Sanskruti Index (neighborhood degree based topological index of Augmented Zagreb Index) shows higher value(|0.9553|) than Augmented Zagreb Index. In the same way, ABC₄ Index, NH index, NM₂ index, NR index &NSUM index express higher correlation with entropy of n-octane isomers than their corresponding degree-based topological indices.

Acentric Factor This propertyof n-octane isomers is highly correlated with neighborhood degree-sum of the above-mentioned topological indices than their corresponding degree- based indices except neighborhood version of Hyper Zagreb Index & Second Zagreb Index (considerably insignificant deviation).

HVAP&DHVAP This correlation coefficients linking the HVAP&DHVAP of n-octane isomers and mentioned degree based topological indices exhibit higher value than their corresponding neighborhood version of degree-based indices. It indicates that, degree-based topological indices should be preferred in designing quantitative structure-property relations with the above Physico-chemical properties of n-octane isomers.

(b) Generalized formulation of Augmented Zagreb Atom Bond Connectivity Index (AzABC)α(G) = ^{𝑑ᵤ 𝑑ᵥ}

𝑑ᵤ+𝑑ᵥ−2 𝛼

𝑢,𝑣 ∊𝐸 where α𝜖R

When α = 3, Az(G) = ^{𝑑ᵤ 𝑑ᵥ}

𝑑ᵤ+𝑑ᵥ−2 3

𝑢,𝑣 ∊𝐸 and α = -1/2, ABC(G) = ^{𝑑ᵤ+𝑑ᵥ−2}

𝑑ᵤ 𝑑ᵥ 𝑢,𝑣𝜖𝐸

The interesting factor of these two indices is that they show higher correlation value than other degree-based indices with boiling point of n-octane isomers. This feature leads us to investigate with numerous values of α to get comparatively good correlation for further studies.

Table 5. shows correlation coefficients of boiling point of n-octane isomers with the indices.

α |(AzABC)_α(G)|

1 0.887 2 0.9042

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http://annalsofrscb.ro 4487 3 0.9223 Az(G)

-1/3 0.8656

-1/2 0.8631 ABC(G) -1 0.8564

-2 0.8434

From the results of Table 5 reveal that, when α is positive, the correlation of boiling point of n-octane isomers high with increasing α values and at the same time α is negative, the correlation decreased with decreasing α values.

Conclusion

From the above results, we conclude that degree-based topological indices are more suitable than neighborhood degree-sum topological indices in the predictive study of boiling points of structural isomers. Among the indices Augmented Zagreb Index and Atom Bond Connectivity Index are suitable for structure property relations linked with boiling points of n-octane isomers. HVAP and DHVAP of isomers are highly correlated with neighborhood degree-sum topological indices. This paper will pave the path for many intricate predictive analyses of structural isomers for their chemical applications.

Table 6. Shows the of experimental values[18] of boiling point, entropy, acentric factor, enthalpy of vaporization & standard enthalpy of vaporization of eighteen structural

octane isomers.

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