Fifth Geometric--Arithmetic Index VS Atom--Bond Connectivity Index and Heat of Formation

TL;DR

This paper explores the correlation between the fifth geometric--arithmetic index and other topological indices, demonstrating its potential in predicting heat of formation for hydrocarbons.

Contribution

It introduces the analysis of the fifth geometric--arithmetic index's predictive power, especially its strong correlation with the atom--bond connectivity index and heat of formation.

Findings

Strong correlation with atom--bond connectivity index for hydrocarbons

Good correlation with heat of formation

Potential for improved molecular property prediction

Abstract

The geometric--arithmetic indices are widely considered in the chemical graph theory in the last decade. The reason of introducing new indices is to gain prediction of target properties of considered molecules that is better than the prediction obtained by already known indices. In the case of the fifth geometric--arithmetic index hitherto no ability of prediction of some molecule property has been considered. In this paper we investigate correlations between the fifth geometric--arithmetic index and some other degree--based topological indices on the family of octane isomers and polyaromatic hydrocarbons. Since a very good correlation is established with the well known atom--bond connectivity index for polyaromatic hydrocarbons and then for the alkane series, the relation between the heat of formation and the fifth geometric--arithmetic index is examined and a good correlation is confirmed in that case as well.

Tables

Table 1. Table 1: The regression statistics between the fifth geometric–arithmetic index and other degree–based topological indices.

Index	Multiple R	R Square	Adjusted R Square	Standard Error
Atom bond connectivity I (ABC)	0,9997	0,9994	0,9994	0,1621
1. ZG valence I (ZM1V)	0,9984	0,9967	0,9967	0,3730
Randić connectivity I ($\chi$)	0,9982	0,9964	0,9963	0,3928
Narumi simple I (Snar)	0,9982	0,9964	0,9964	0,3893
Pogliani I (Dz)	0,9974	0,9949	0,9948	0,4670
First Zagreb I (ZM1)	0,9975	0,9950	0,9949	0,4589
2. ZG valence I (ZM2V)	0,9964	0,9928	0,9927	0,5549
Second Zagreb I (ZM2)	0,9938	0,9877	0,9876	0,7232
Harary I (Har)	0,9972	0,9944	0,9944	0,4866
Quadratic I (Q)	0,9813	0,9630	0,9625	1,2564
Ramification I (Ram)	0,9670	0,9352	0,9343	1,6628
Total structure connectivity I (Xt)	0,9348	0,8739	0,8724	2,3184
Schultz MTI (SMTI)	0,9130	0,8335	0,8315	2,6641
Gutman MTI (GMTI)	0,9066	0,8219	0,8196	2,7559

Table 2. Table 2: The fifth geometric–arithmetic index and atom–bond connectivity index for PAH’s (part 1).

Molecule	$G{A}_5$	$ABC$	Molecule	$G{A}_5$	$ABC$
1. naphtalene	10,9193	7,73773	31. 4-5-methylenephenanthrene	17,854	12,5257
2. 1-methylnaphthalene	11,8465	8,51379	32. tetracene	20,8956	14,7279
3. 2-methylnaphthalene	11,9068	8,55423	33. benzo[a]anthracene	20,8547	14,6875
4. 1-ethylnaphthalene	12,8067	9,11151	34. chrysene	20,8069	14,647
5. 2-ethylnaphthalene	12,8612	9,15195	35. benzo[c]phenanthrene	20,8216	14,647
6. 2-6-dimethylnaphthalene	12,8943	9,37073	36. triphenylene	20,8009	14,6066
7. 2-7-dimethylnaphthalene	12,8943	9,37073	37. pyrene	18,832	13,2328
8. 1-7-dimethylnaphthalene	12,8397	9,33029	38. 1-methylpyrene	19,7799	14,0089
9. 1-5-dimethylnaphthalene	12,7781	9,28985	39. 2-methylpyrene	19,8264	14,0493
10. 1-2-dimethylnaphthalene	12,7804	9,28985	40. 4-methylpyrene	19,7826	14,0089
11. 1-3-7-trimethylnaphthalene	13,8272	10,1468	41. 2-7-dimethylpyrene	20,8208	14,8658
12. 2-3-5-trimethylnaphthalene	13,7722	10,1063	42. pentacene	25,8837	18,223
13. 2-3-6-trimethylnaphthalene	13,8268	10,1468	43. dibenzo[ai]anthracene	25,8429	18,1826
14. phenalene	14,8554	10,4853	44. dibenzo[ah]anthracene	25,802	18,1421
15. 1-phenylnaphthalene	17,8485	12,6066	45. dibenzo[aj]anthracene	25,802	18,1421
16. 2-phenylnaphthalene	17,8854	12,647	46. benzo[b]chrysene	25,8007	18,1421
17. anthracene	15,9074	11,2328	47. dibenzo[ac]anthracene	25,8005	18,1017
18. 1-methylanthracene	16,8403	12,0089	48. pycene	25,7529	18,1017
19. 2-methylanthracene	16,8949	12,0493	49. benzo[a]pyrene	23,8039	16,6875
20. 2-7-dimethylanthracene	17,2661	12,567	50. benzo[e]pyrene	23,7798	16,647
21. 2-6-dimethylanthracene	17,8825	12,8658	51. perylene	23,7639	16,647
22. 2-3-dimethylanthracene	17,8275	12,8254	52. coronene	27,7425	19,282
23. 9-10-dimethylanthracene	17,6914	12,7041	53. anthranthrene	26,8044	18,7279
24. phenanthrene	15,8609	11,1924	54. benzo[ghi]perylene	26,7745	18,6875
25. 1-methylphenanthrene	16,7925	11,9684	55. dibenzo[ae]pyrene	28,7574	20,1017
26. 2-methylphenanthrene	16,8484	12,0089	56. 1-methylchrysene	21,7594	15,4231
27. 3-methylphenanthrene	16,8541	12,0089	57. 6-methylchrysene	21,2586	15,0006
28. 4-methylphenanthrene	16,799	11,9684	58. 3-methylcholanthrene	24,7406	17,4635
29. 9-methylphenanthrene	16,7945	11,9684	59. indeno[1-2-3-cd]pyrene	26,7853	18,6875
30. 3-6-dimethylphenanthrene	17,8473	12,8254	60. pentaphene	25,8486	18,1826

Table 3. Table 3: The fifth geometric–arithmetic index and atom–bond connectivity index for PAH’s (part 2).

Molecule	$G{A}_5$	$ABC$
61. hexaphene	30,8367	21,6777
62. indano	9,91929	7,03063
63. indene	9,91929	7,03063
64. azulene	10,9193	7,73773
65. acenaphthene	13,8678	9,77817
66. acenaphthylene	13,8678	9,77817
67. fluorene	14,8829	10,4853
68. 1-methylfluorene	15,8202	11,2613
69. 2-methylfluorene	15,8704	11,3018
70. 3-methylfluorene	15,8761	11,3018
71. 4-methylfluorene	15,3347	10,8388
72. 9-methylfluorene	15,7793	11,2209
73. 1-2-benzofluorene	19,8346	13,9399
74. fluoranthene	18,8156	13,1924
75. 2-3-benzofluorene	19,8768	13,9804
76. 3-4-benzofluorene	19,8436	13,9399
77. benzo[ghi]fluoranthene	21,8103	15,2328
78. benzo[k]fluoranthene	23,8152	16,6875
79. benzo[b]fluoranthene	23,7933	16,647
80. benzo[j]fluoranthene	23,7802	16,647
81. ovalene	40,7576	28,223
82. quaterryllene	49,5342	34,4657

Table 4. Table 4: Heat of formation and the fifth geometric–arithmetic index and of some PAH’s and the alkane series.

Molecule	heat of formation	$G{A}_5$	Molecule	heat of formation	$G{A}_5$
	$\Delta H$ $\left[kJ/mol\right]$			$\Delta H$ $\left[kJ/mol\right]$
naphtalene	150,6	10,9193	ethane	20.4	1,000
anthracene	227,7	15,9074	propane	25,02	2,000
phenanthrene	207,1	15,8609	butane	30,03	2,6200
benzo[a]anthracene	291	20,8547	pentane	35,08	3,9391
chrysene	262,8	20,8069	hexane	39,96	4,9391
benzo[c]phenanthrene	302,4	20,8216	heptane	44,89	5,9391
triphenylene	269,8	20,8009	octane	49,82	6,9391
pyrene	225,7	18,832	nonane	54,66	7,9391
pentacene	374	25,8837	decane	59,67	8,9391
dibenzo[ah]anthracene	343	25,802	undecane	64,61	9,9391
dibenzo[aj]anthracene	343	25,802	dodecane	69,52	10,9391
benzo[b]chrysene	346	25,8007	tridecane	74,45	11,9391
dibenzo[ac]anthracene	345	25,8005	tetradecane	79,38	12,9391
pycene	334	25,7529	pentadecane	84,30	13,9391
benzo[a]pyrene	301	23,8039	hexadecane	89,22	14,9391
benzo[e]pyrene	304	23,7798	heptadecane	94,2 17	15,9391
perylene	324	23,7639	octadecane	99,08	16,9391
pentaphene	359	25,8486	nonadecane	104,00	17,9391
			icosane	108,93	18,9391

Equations

$$G{A}_{gen}(G)=\sum _{uv\in E(G)}\frac{2\sqrt{Q_u{Q}_v}}{Q_u+Q_v},$$

$$G{A}_5(G)=\sum _{uv\in E(G)}\frac{2\sqrt{S_u{S}_v}}{S_u+S_v},$$

$${\text{GA}}_5(G)$$

$$\approx 11,8465.$$

$$Z{M}_1(G)=\sum _{v\in V(G)}{d}^2(v)$$

$$Z{M}_2(G)=\sum _{uv\in E(G)}d(u)d(v).$$

$$Z{M}_1^v(G)=\sum _{u\in V(G)}(d^v(u){)}^2\text{and}Z{M}_2^v(G)=\sum _{uw\in E(G)}(d^v(u){)}^2(d^v(w){)}^2.$$

$$\chi (G)=\sum _{uv\in E(G)}\frac{1}{\sqrt{d(u)d(v)}},$$

$$Dz(G)=\sum _{u\in V(G)}{d}^Z(u),$$

$$ABC(G)=\sum _{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}},$$

$$Ram(G)=\sum _{v\in V(G),d(v)\ge 3}(d(v)-2),$$

$$Snar(G)=\prod _{v\in V(G)}d(v),$$

$$Xt(G)=\prod _{v\in V(G)}\frac{1}{\sqrt{d(v)}},$$

$$Q(G)=\frac{1}{2}\sum _g((g^2-g){}^{g}F+2)=3-2|V(G)|+\frac{M_1(G)}{2},$$

$$\begin{array}[]{rcl}GA_{5}(P_{n})&=&2(\frac{\sqrt{2\cdot 3}}{5}+\frac{\sqrt{3\cdot 4}}{7}+(n-5)\frac{\sqrt{4\cdot 4}}{8}+\frac{\sqrt{3\cdot 4}}{7}+\frac{\sqrt{2\cdot 3}}{5})\\ &=&4\frac{\sqrt{6}}{5}+8\frac{\sqrt{3}}{7}+n-5\\ ABC(P_{n})&=&(n-1)\frac{\sqrt{2}}{2}\end{array}$$

$$ABC(P_n)=0.7071G{A}_5(P_n)+0.0431.$$

Full text

In this paper we investigate correlations between the fifth geometric–arithmetic index and some other degree–based topological indices on the family of octane isomers and polyaromatic hydrocarbons. Since a very good correlation is established with the well known atom–bond connectivity index for polyaromatic hydrocarbons and then for the alkane series, the relation between the heat of formation and the fifth geometric–arithmetic index is examined and a good correlation is confirmed in that case as well.

Keywords: Fifth geometric–arithmetic index, atom–bond connectivity index, heat of formation, correlation.

INTRODUCTION

Among the molecular descriptors which provide information on molecular constitution, topological indices have several assets such as: (1) easy calculation with very low computer time (cpu) requirements; (2) diversity of possibilities to choose from in order to match properties of the data set; (3) high correlation ability with chemical and physical properties or biological activities. There are numerous of topological indices that have found some applications in theoretical chemistry, especially in QSPR/QSAR research. Within all topological indices ones of the most investigated are the descriptors based on the valences of atoms in molecules, so-called degree–based topological indices.

Among the degree–based topological indices a class of geometric–arithmetic topological indices 18 may be defined as

[TABLE]

where $Q_{u}$ is some quantity that can be in a unique manner associated with vertex $u$ of graph $G$. The first member of this class was considered by Vukičević and Furtula 33 in year 2009 by setting $Q_{u}$ to be the vertex degree $d(u)$. One year later Fath-Tabar et al. 16 introduced the second such index by setting $Q_{u}$ to be the number $n_{u}$, which is the number of vertices of G lying closer to vertex $u$ than to vertex $v$ for edge $uv$ of a graph $G$. The edge variant was studied by Bo Zhou et al. 35 in 2009 and lead to the third geometric–arithmetic index; $Q_{u}$ being the number $m_{u}$ of edges of $G$ lying closer to vertex $u$ than to vertex $v$ for edge $uv$ of a graph $G$. The fourth member of this class was considered by Ghorbani et al. 21 in 2010 by setting $Q_{u}$ to be the eccentricity of vertex $u$ denoted by $\varepsilon_{u}$ and finally the fifth geometric–arithmetic index was defined in 2011 by Graovac et al. 22 by letting $Q_{u}$ to be the sum of degrees over all vertices incident with vertex $u$.

Beside that the edge and the total versions of geometric–arithmetic indices were considered26, 25 and recently Wilczek37 defined nine new geometric–arithmetic indices. Some mathematical properties of first four geometric–arithmetic indices are obtained in 5, 6, 7, 8, 9, 10, 31, 32. It was also shown that first three geometric–arithmetic indices possess relatively good descriptive as well as predictive capabilities with respect to some selected properties of octanes and benzenoid hydrocarbons8, 33.

After the introduction of the fifth geometric–arithmetic index it has been calculated for many different families of (molecular) graphs, but there were no known correlation nor with physico-chemical properties of molecules nor with any other topological indices. The aim of the present paper is to fill this gap. The fifth geometric–arithmetic index is compared with some other distance-based topological indices and physico-chemical properties. As it is best correlated with the atom–bond connectivity index, which is used for predicting the heat of formation of certain hydrocarbons, the connection between the fifth geometric–arithmetic index and the heat of formation is established.

THE FIFTH GEOMETRIC-ARITHMETIC INDEX AND SOME OTHER DISTANCE-BASED TOPOLOGICAL INDICES

A graph $G$ is an ordered pair $G=(V,E)$ of a set $V$ of vertices (also called nodes or points) together with a set $E$ of edges, which are $2$-element subsets of $V$ (more information about some basic concepts in graph theory can be found in a book written by West36). Having a molecule, if we represent atoms by vertices and bonds by edges, we obtain a molecular graph.

The graphs considered in this paper are all finite and connected. The degree $d(u)$ of a vertex $u\in V(G)$ is the number of edges incident to vertex $u$.

The fifth geometric–arithmetic index is defined as

[TABLE]

where $\displaystyle S_{u}=\sum_{uv\in E(G)}d(v)$.

As an example, we calculate the fifth geometric–arithmetic index for 1-methylnaphthalene (see Figure 1).

First we denote the molecular graph of 1-methylnaphthalene by $G$. Then

[TABLE]

The fifth geometric–arithmetic index was firstly computed for nanostar dendrimers 22, followed by circumcoronene series12, zig-zag polyhex nanotubes and nanostars13, $TURC4C8(S)$ nanotube14, armchair polyhex nanotubes15 and polyaromatic hydrocarbons1. Beside that this index was computed for naphtalenic nanosheet $[4n,2m]$3, fan and wheel molecular graphs19, bridge and carbon nanocones20 and just recently for para-line graphs in convex polytopes17.

Probably the most studied degree–based topological indices are the Zagreb indices which have been introduced almost fifty years ago by Gutman and Trinajstič 24. The first Zagreb index is defined as

[TABLE]

and the second Zagreb index equals

[TABLE]

Zagreb indices can also be calculated by using the valence vertex degree $d^{v}(u)$ (i.e. the number of valence electrons of $u$ minus the number of hydrogen atoms attached to $u$) resulting in the first and the second valence Zagreb indices

[TABLE]

Some other important degree–based topological indices are Randić connectivity index 30

[TABLE]

the Pogliani index 29

[TABLE]

where $d^{Z}(u)$ is the $Z$-delta number of $u$ and is defined as a quotient between the number of valence electrons and the principal quantum number of vertex $u$. Further we have the atom–bond connectivity index 11

[TABLE]

the ramification index 2

[TABLE]

Narumi simple index 27

[TABLE]

the total structure connectivity index 28

[TABLE]

and the quadratic index 4, also called normalized quadratic index

[TABLE]

where $g$ are the different vertex degree values and ${}^{g}F$ is the vertex degree count.

COMPUTATIONAL DETAILS

In the present section we give an algorithm which we use to compute the fifth geometric–arithmetic index. The algorithm contains two special functions, i.e. CalculateDegree and CalculateS.

Let $G$ be a graph given by an adjacency matrix with vertices $1,2,\ldots,n$. The function CalculateDegree computes degree for a given vertex. The function CalculateS determines the value which is the sum of degrees of all neighbors for a given vertex. Note that this two special functions can be obtained easily and both functions have the time complexity $O(n^{2})$.

The fifth geometric–arithmetic indices for the polyaromatic hydrocarbons molecules are collected in Tables 2 and 3; and for the alkane series in Table 4.

Since the data for the atom–bond connectivity index for the polyaromatic hydrocarbons is not available, we compute these values with a simple algorithm. The algorithm for calculating the atom–bond connectivity index is quite similar to the algorithm for calculating the fifth geometric–arithmetic index. For the completeness of the paper, we give it anyway.

As before, let $G$ be a graph given by an adjacency matrix with vertices $1,2,\ldots,n$ and the function CalculateDegree computes degree for a given vertex. Let us mention that Algorithm 2 has the time complexity $O(n^{2})$.

The atom–bond connectivity indices for the polyaromatic hydrocarbons are collected in Tables 2 and 3.

RESULTS AND DISCUSSION

A benchmark data set for the octane isomers and the polyaromatic haydrocarbons is available at www.moleculardescriptors.eu.

For the set of 18 octane isomers we have compared the fifth geometric–arithmetic index with 16 physico-chemical properties and then further with the degree–based topological indices - unfortunately no significant correlation could be established.

Next we perform the regression analysis on the set of 82 polyaromatic hydrocarbons. The benchmark data set enables the analysis of three physico-chemical properties (melting and boiling point, octanol-water partition coefficient); we could not establish any correlation between the fifth geometric–arithmetic index and these properties. Further we consider degree–based indices and it turns out that the best correlation is the correlation between the fifth geometric–arithmetic index and the atom–bond connectivity index (see Figure 2). More precisely, the regression statistics for these two indices is: multiple $R$ is $0,9997$, $R^{2}$ is= $0,9994$; adjusted $R^{2}$ is 0,9994 and standard error is 0,1621. Good correlation probably follows from the fact that the formulas for computing the fifth geometric–arithmetic index and the atom–bond connectivity index are quite similar.

The second best correlation is the correlation between the fifth geometric–arithmetic index and the first valence Zagreb index. In this case, the regression statistics is as follows: multiple $R$ is 0,9984, $R^{2}$ is 0,9967; adjusted $R^{2}$ is 0,9967 and standard error is 0,3730. The correlation between the fifth geometric–arithmetic index and the Randić connectivity index is also very good. More precisely, multiple $R$ is 0,9982; $R^{2}$ is 0,9964; adjusted $R^{2}$ is 0,9963 and standard error is 0,3928. The correlation between the fifth geometric–arithmetic index is also very high with the following indices: Narumi simple index, Pogliani index, the first and the second Zagreb index, the second Zagreb valence index, and Harary index. One can see that the regression statistics for these indices is: multiple $R$ greater than 0,993, $R^{2}$ is greater than 0,987; adjusted $R^{2}$ is greater that 0,987 and standard error is at most 0,724. The quadratic index, the ramification index, the total structure connectivity index also have a good correlation with the fifth geometric–arithmetic index. In this case the standard error is greater than 1, $R^{2}$ is between 0,94 and 0,98, multiple $R$ and adjusted $R^{2}$ are both between 0,87 and 0,96. From this follows that correlation is still good, but in respect to other indices it is relatively poor. In the case of Schultz index and Gutman index the correlation is weaker. The regression statistics between the fifth geometric–arithmetic index and the degree–based topological indices mentioned here are all collected in Table 1.

As we can see the fifth geometric–arithmetic index correlates the best with the atom–bond connectivity index. The atom–bond connectivity index was introduced in 1989 by Estrada et al.11 as a tool to describe the heat of formation $\Delta H$ of alkanes since it has shown a good quantitative structure-property relationship (QSPR) model. We have therefore used the available data 34 and checked if there is any correlation between the fifth geometric–arithmetic index and the heat of formation of certain polyaromatic hydrocarbons. The data gathered in Table 4 results in the linear regression with the multiple $R$ being 0,971 and $R^{2}$ equals 0,942, which is similar to the correlation obtained by by Gutman and Furtula 23, where the heat of formation of some polyaromatic hydrocarbons was compared with the first geometric–arithmetic index and the established correlation coefficient was 0,972.

Since the seminal paper 11 on atom–bond connectivity index models the heat of formation $\Delta H$ of alkanes, our next aim is to compare $\Delta H$ of alkanes with theirs fifth geometric–arithmetic index. The data in Table 4 is taken form www.webbook.nist.gov and results in a linear regression with the multiple $R=0,9999$ and $R^{2}=0,9998$. The good correlation is due to a linear relation between the fifth geometric–arithmetic index and the atom–bond connectivity index of the alkane series. The molecular graph of an alkane $C_{n}H_{2n}$ is a path $P_{n}$ on $n$ vertices, so for $n\geq 5$ a straightforward calculation yields in

[TABLE]

what gives

[TABLE]

CONCLUSIONS

The fifth geometric–arithmetic index is an index in the family of the degree–based topological indices. The index is relatively new and although its mathematical properties and closed formulae for some families of chemical graphs were derived, there is no proven relation between the fifth geometric–arithmetic index and physico-chemical properties or some other (degree–based) topological indices so far.

In this paper we consider three types of molecules. In the case of octane isomers no significant results could be shown. For the polyaromatic hydrocarbons and the alkane series very good correlation and linear relation, respectively, between the fifth geometric–arithmetic index and the atom–bond connectivity index is established. As a consequence the fifth geometric–arithmetic index is related with the heat of formation. Our data set in that case consisted of 18 polyaromatic hydrocarbons and 19 members of the alkane series, what is not enough for a credible QSPR analysis and this is a problem which could be considered in the future by the chemical society, since the Algorithm 1 enables the calculation of the fifth geometric–arithmetic index.

ACKNOWLEDGMENTS

The authors Matevž Črepnjak and Petra Žigert Pleteršek acknowledge the financial support from the Slovenian Research Agency, research core funding No. P1-0403 and No. P1-0297, J1-9109, respectively.