# Comparisons of the Sombor Index of Alkane, Alkyl, and Annulene Series with Their Molecular Mass

Authors: Maryam Mohammadi; Hasan Barzegar; Ali Reza Ashrafi
Journal: Journal of Chemistry
 (2022)

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## Abstract

Suppose G is an undirected simple graph. A topological index for G is a number with this property that it is invariant under all graph isomorphisms with applications in chemistry. The Sombor and reduced Sombor indices of G are defined asSOG=∑uv∈EGdG2u+dG2v and SOredG=∑uv∈EGdGu−12+dGv−12, respectively. Here, dGu denotes the degree of the vertex u in G. In this paper, these invariants were computed for alkanes, alkyls, and annulenes. A comparison of our calculations with molecular mass is also presented. As a consequence, it is shown that there is a good correlation between Sombor index and molecular mass of these compounds.


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## Body

## 1. Introduction

Suppose M is a molecule. The molecular graph of M is a graph with the set of all atoms as its vertex set and chemical bonds are the edges of this graph. We use the notation G (M), G for short, for this graph. For each vertexu,dGu denotes the degree of u. It is usual to use the notation VG for the vertex set of G. Note that in molecular graphs, the degree of each vertex is assumed to be at most 4.Gutman [1] proposed the degree-based topological index “Sombor index” as SOG=∑uv∈EGdG2u+dG2v. In the mentioned paper, Gutman proved that the complete graph Kn has the maximum value of Sombor index in the set of all n-vertex graphs and the star graph Sn has the maximum value of Sombor index in the set of all n-vertex trees. The minimum values of Sombor index in the set of all n-vertex graphs and n-vertex trees are the n-vertex null graph ∅n and the n-vertex path graph Pn, respectively. We refer to [2, 3] and references therein for more information on this topic.Alkanes are acyclic saturated hydrocarbons. They are important chemical structures consisting of carbon and hydrogen in a tree structure with the general chemical formula CnH2n+2. Alkyls are another chemical compounds with the same materials and general formula CnH2n+1. Annulenes are monocyclic hydrocarbons with the general chemical formula CnHn+1 or CnHn, when n is odd or even, respectively. We refer to [4, 5] for more information on the importance of these compounds on chemistry.The atomic mass of an atom is defined as the mass of this atom. It is calculated based on a single carbon-12 atom. It is usually expressed in terms of the unified mass (u). From the periodic table [4, 5], we know that the atomic mass of carbon is 12.0107u and that for hydrogen is 1.00794u. Furthermore, the molecular mass M for a molecule X is defined as the sum of the atomic masses of all atoms in X.Redžepović [6] examined the predictive and discriminative potentials of the Sombor and reduced Sombor indices of chemical graphs and showed that these molecular descriptors have good predictive potential. In a recent paper [7], Liu et al. reviewed the existing bounds and extremal results related to the Sombor index and its variants.


## 2. Main Results and Discussion

The alkanes have the general formulaCnH2n+2 [4]. The structural formulas of the first four members of the chain of alkanes are depicted in Figure 1.Figure 1
Straight chain of some alkanes.We also have a branched chain of alkanes like isobutane or 2-methylpropane with the graph structure depicted in Figure2.Figure 2
Branched chain of butane (isobutane) and its condensed structural formulas.By our discussion about molecular mass on the last paragraph of Section1, the atomic mass of alkanes can be calculated by MCnH2n+2=14.026n+2.016 where n is the number of carbon atoms in the alkane under consideration.Theorem 1.
The Sombor and reduced Sombor indices of the alkanes withn carbon atoms can be calculated by the following formulas:(1)SOCnH2n+2=42+217n+217−42,SOredCnH2n+2=32+6n+6−32.Proof.
Define an equivalence relation on the straight and branched-chain alkanes and then apply mathematical induction.
The absolute error of a calculation is defined as the amount of error in measurements. It is the difference between the measured and actual values. In what follows, we compare the Sombor index of alkanes with their molecular masses. To do this, its absolute error will be calculated.Theorem 2.
Ifan=MCnH2n+2−SOCnH2n+2, then the absolute error can be evaluated as an=MCnH2n+2−SOCnH2n+2≈0.123n−0.573.Note thatlimn⟶+∞MCnH2n+2−SOCnH2n+2≈limn⟶+∞0.123n−0.573=+∞.Theorem 3.
0.972<MCnH2n+2/SOCnH2n+2<1.009.Proof.
It is easy to calculate thatMCH4/SOCH4≈0.9727, and so 0.972<MCnH2n+2/SOCnH2n+2. Other terms of the sequence an=MCnH2n+2/SOCnH2n+2 are as follows:(2)0.973,0.989,0.995,0.999,1.001,1.002,…,14.026n+2.01642+217n+217−42.
The limit of the sequence is approximately1.009 because(3)limn⟶+∞an=limn⟶+∞14.026n+2.01642+217n+217−42=limn⟶+∞14.026n42+217n=14.02642+217≈1.0088≈1.009.
This proves thatMCnHn+2/SOCnHn+2<1.009, and none of the sequence sentences are equal to the sequence limit. So, 0.972<MCnH2n+2/SOCnH2n+2<1.009.
The reduced Sombor indexSOredG and the Sombor index SO(G) of the first nineteen alkanes are shown in Table 1.
As shown in the graph of Figure3, the equivalent values of the Sombor index and molecular mass, based on the number of carbons, are approximately equal or close to each other. Also, Table2 shows that after the fourth alkane, the molecular mass is more than the Sombor index.
If a hydrogen atom is removed from an end carbon atom of a straight-chain alkane, then the resulting compound is called alkyl. For example, removing a hydrogen atom from methane gives the methyl group, OCH3 [4]. The general formula of the straight-chain and the branched-chain alkyls is CnH2n+1, and it is clear that the molecular mass general formula is computed as MCnHn+1=14.026n+1.008, where n is the number of the carbon in the alkane.Table 1
Sombor index and molecular mass of some alkanes.
Name of alkaneSOredGSO (G)Molecular mass (g/mol)Mass/SoGMethane1216.49216.0420.973Ethane22.24230.39530.0680.989Propane32.48544.29944.0940.995Butane42.72858.20258.1200.999Pentane52.97172.10572.1461.001Hexane63.21386.00886.1721.002Heptane73.45699.911100.1981.003Octane83.698113.814114.2241.004Nonane93.941127.717128.2501.004Decane104.184141.620142.2761.005Undecane114.426155.523156.3021.005Dodecane124.669169.426170.3281.005Tridecane134.912183.329184.3541.006Tetradecane145.154197.232198.3801.006Pentadecane155.397211.135212.4061.006Hexadecane165.604225.038226.4321.006Heptadecane175.882238.941240.4581.006Octadecane186.125252.845254.4841.006Nonadecane196.368266.748268.511.007Figure 3
Sombor index and molecular mass of the first nineteen alkanes.Table 2
Sombor index and molecular mass of some annulenes (n=odd≥3).
Name of annuleneSOredGSO (G)Molecular mass (g/mol)Mass/SOGn=odd[3]-Annulene24.72833.46340.0621.197[5]-Annulene39.21353.02366.0981.246[7]-Annulene53.69872.58392.1341.269[9]-Annulene68.18492.143118.1701.282[11]-Annulene82.669111.703144.2061.290[13]-Annulene97.154131.263170.2421.296[15]-Annulene111.640150.823196.2781.301[17]-Annulene126.125170.382222.3141.304[19]-Annulene140.610189.942248.3501.307Theorem 4.
The Sombor and reduced Sombor indices of the alkyls are computed from the following formulas:(4)SOCnHn+1=310,n=1,217+42n+210−17−82+5,n≥2,SOredCnHn+1=6,n=1,6+32n+1−62+13,n≥2.Proof.
The result can be proved by defining an equivalence relation on straight and branched chain of alkyls and then applying the mathematical induction onn.Theorem 5.
1.009<MCnH2n+1/SOCnH2n+1<1.584.Proof.
It is easy to see thatMCH3/SOCH3≈1.5846, and so 1.548>MCnH2n+1/SOCnH2n+1. Consider the sequence with general formula an=MCnH2n+1/SOCnH2n+1. Then, approximately a1=1.585, and for n≥2, the terms of the sequence are as follows:(5)1.227,1.146,1.109,1.088,1.074,1.064,…,14.026n+1.008217+42n+210−17−82+5.
The limit of this sequence is approximately1.009 because(6)limn⟶+∞an≈limn⟶+∞14.026n+1.008217+42n+210−17−82+5=limn⟶+∞14.26n217+42n=14.026217+42≈1.0088≈1.009.
Since alwaysMCnH2n+1/SOCnH2n+1>1.009, none of the sequence terms are equal to the sequence limit. Therefore, 1.009<MCnH2n+1/SOCnH2n+1<1.584.
Table3 displays the Sombor index and molecular mass values of the first nineteen alkyls. In Figure 4, the upper line shows the molecular mass and the lower line is for the Sombor index.
Annulene belongs to a series of conjugated monocyclic hydrocarbons with chemical formulasCnHn, where n is even, and CnHn+1, where n is odd. Here, n denotes the number of carbon atoms. Some of the most important members of this series are benzene ([6]-annulene), cyclobutadiene ([4]-annulene), cyclooctatetraene ([8]-annulene), and cyclotetradecaheptaene ([14]-annulene) [8]. It is clear that the molecular mass formula for annulenes with formula CnHn+1n=odd≥3 is MCnHn+1=13.018n+1.008, and that for the annulenes with formula CnHnn=even≥4 is MCnHn=13.018n. Tables 3 and 4 show the values of the Sombor index and molecular mass of some annulenes. Their graphs are also drawn in Figures 5 and 6.Table 3
Sombor index and molecular mass values of some alkyls.
Name of alkylSOredGSO (G)Molecular mass (g/mol)Mass/SOGMethyl69.48715.0341.585Ethyl16.60623.69429.0601.227Propyl26.84837.59743.0861.146Butyl37.09151.50057.1121.109Pentyl47.33365.40371.1381.088Hexyl57.57679.30685.1641.074Heptyl67.81993.20999.1901.064Octyl78.061107.112113.2161.057Nonyl88.304121.015127.2421.052Decyl98.547134.918141.2681.047Undecyl108.789148.821155.2941.044Dodecyl119.032162.725169.3201.041Tridecyl129.275176.628183.3461.038Tetradecyl139.517190.531197.3721.036Pentadecyl149.760204.434211.3981.034Hexadecyl160.003218.337225.4241.033Heptadecyl170.245232.240239.4501.031Octadecyl180.488246.143253.4761.030Nonadecyl190.730260.046267.5021.029Figure 4
The Sombor index and molecular mass of nineteen alkyls with the correlation coefficient 1.Table 4
Sombor index and molecular mass of some annulenes (n=even≥4).
Name of annuleneSOredGSO (G)Molecular Mass (gr/mol)Mass/SOGn=even[4]-Annulene28.97139.12052.0721.331[6]-Annulene43.45658.68078.1081.331[8]-Annulene57.94178.240104.1441.331[10]-Annulene72.42697.800130.1801.331[12]-Annulene86.912117.360156.2161.331[14]-Annulene101.397136.919182.2521.331[16]-Annulene115.882156.479208.2881.331[18]-Annulene130.368176.039234.3241.331Figure 5
Diagram of Sombor index and molecular mass for some annulenes with oddn.Figure 6
Diagram of Sombor index and molecular mass for some annulenes (n=even≥4).Theorem 6.
The Sombor and reduced Sombor indices of the annulenes are computed from the following formulas:(7)SOG=82+217m+42+217m=n−12,n=odd≥3,82+217m+82+217m=n−22,n=even≥4,SOredG=6+62m+6+32m=n−12,n=odd≥3,6+62m+6+62m=n−22,n=even≥4.Proof.
Induct onm.Theorem 7.
1.197<MCnHn+1/SOCnHn+1<1.331.Proof.
It is easy to calculate thatMC3H4/SOC3H4≈1.1972, and so 1.197<MC3H4/SOC3H4. Consider the sequence an=MCnHn+1/SOCnHn+1,n=odd≥3. The terms of this sequence are(8)1.197,1.245,1.269,1.282,1.290,1.296,…,13.018n+1.00882+217m+42+217,which shows that(9)limn⟶+∞an=limn⟶+∞13.018n+1.00882+217n−1/2+42+217=limn⟶+∞13.018n82+217/2n=13.01882+217/2≈1.331.
The limit is 1.331, and hence1.197<MCnHn+1/SOCnHn+1<1.331. Similarly, we define bn=MCnHn/SOCnHn,n=even≥4. Then, the terms of this sequence are(10)1.331,1.331,1.331,1.331,1.331,1.331,…,13.018n82+217n−2/2+82+217,and so(11)limn⟶+∞an=limn⟶+∞13.018n82+217n−2/2+82+217=limn⟶+∞13.018n82+217/2n=13.01882+217/2n≈1.331.
This completes our argument.Theorem 8.
The Sombor index of the annulene seriesn=even≥4 can be approximately written as MCnHn≈1.331×SOCnHn.Proof.
Induct onn.


## 3. Conclusion

In this article, the Sombor and reduced Sombor indices of the alkane, alkyl, and annulene groups were presented. We compare these values with the molecular mass of the first nineteen elements of these groups. It is shown that the fractionMG/SOG has a good behavior which shows that Sombor index has good correlation with molecular mass. Finally, for any vertex degree-based topological index TI and/or any homologous series of molecules, the relation between TI and molecular mass is strictly linear.



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