# Degree-Based Topological Indices of Boron B12

Authors: Nouman Saeed; Kai Long; Zeeshan Saleem Mufti; Hafsa Sajid; Abdul Rehman
Journal: Journal of Chemistry
 (2021)

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## Abstract

Topological index sometimes called molecular descriptor is a numerical value which associates a chemical composition for correlating chemical structure with numerous physical properties, chemical reactivity, or biological activity. In this paper, we study some topological indices of boron and try to correlate the physicochemical properties such as freezing points, boiling points, melting points, infrared spectrum, electronic parameters, viscosity, and density of chemical graphs. We discuss these topological indices, and some of them are mentioned here such as Randic index, the first general Zagreb index, the general sum connectivity index, hyper-Zagreb index (HM), the atom-bond connectivity index (ABC), the geometric-arithmetic index (GA), the harmonic index (H), and the forgotten index (F).


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## Body

## 1. Introduction and Applications

A chemical graph theory is an outlet of mathematical chemistry which applied graph theory to the molecular structure of chemical compounds. Topological index is a part of chemical graph theory which correlates the physicochemical properties such as freezing point, boiling point, melting point, infrared spectrum, electronic parameters, viscosity, and density of the underlying chemical graphs. The reader can find bulk of papers on topological indices which have been in print so far [1–5].A graph can be recognized by a polynomial, a matrix, a sequence of numbers, or a numeric number which represents the whole graph, and these representations are designed to be uniquely defined for that graph. Topological indices are major tools for analyzing many physicochemical properties of molecules without performing any testing. Some most significant types of topological indices of graphs are distance-based topological indices, degree-based topological indices, and spectrum-based topological indices. One of the most investigated categories of topological indices used in mathematical chemistry is called degree-based topological indices, which are defined in terms of the degrees of the vertices of a graph.Topological index is a graph invariant which characterizes the topology of the molecular structure and converts the molecular graph into a real number that predicts some physicochemical properties such as freezing point, boiling point, and melting point. Nowadays, a biological testing of chemical compounds is too much expensive. It requires a very large laboratory and advanced equipment to test these compounds. This process is costly and time-consuming. Due to this factor, pharmaceutical companies are very eager to find such new ideas or methods by which cost could be reduced. One can reduce the cost in which no need of laboratories and no need of equipment, but just need to study the certain chemical structure using topological indicesTopological indices are of different types such as degree-based topological indices, distance-based topological indices, and spectrum-based topological indices. The notion of topological index was discovered in 1947 when Harold Wiener was working on the boiling point of paraffin. He named this index as path number. Later on, the path number was renamed as the Wiener index, and then the theory of topological index started. The Wiener index is the first and the most studied topological index. Wiener index is defined as the sum of distances among all the sets of vertices inG [6]. Among all the types of topological indices, degree-based topological indices play an extensive role in chemical graph theory [7]. That motivated us to study the chemical structure of boron B12 under this phenomenon.B12 is a two-dimensional icosahedral network. A recent study of high-pressure solid boron affirmed that the icosahedral B12-containing structures are quite universal [8]. On the other hand, previous experimental and theoretical studies of freestanding boron clusters have shown that the B12 structure is unstable in the gas phase [9, 10]. The (n, m) unit of boron B12 structure is given in the figure, where “n” represents the number of rows and “m” represents the number of columns of the boron B12 structure.LetG be a simple graph with order p and size q. Let dv represent the degree of the vertex v and is defined as the number of edges incident on v. Let Sv represent the sum of the degree of all the neighbors of v [11]. Graph theory is closely connected to the computer science, applied mathematics, optimization theory, web designing, operation research, biology, and chemistry.The general Randic connectivity index ofG is defined as(1)RαG=∑rs∈EGdrdsα,where α represents a real number. If α is −1/2, then R−1/2(G) is said to be the Randic connectivity index of G. Gutman and Trinajstic presented the first general Zagreb index in 1972 [12]:(2)MαG=∑r∈VGdrα.In 2010, the general sum connectivity indexχαG has been invented:(3)χαG=∑rs∈EGdr+dsα.The ABC index was presented by Estrada and Torres et al. in [13]. The ABC index of graph G is expressed as(4)ABCG=∑rs∈EGdr+ds−2drds.Das announced the geometric-arithmetic (GA) index in [14, 15]. The geometric-arithmetic index denoted by GA for graph G is presented by(5)GAG=∑rs∈EG2drdsdr+ds.In 2013, the hyper-Zagreb index has been introduced by Shirdel et al. as(6)HMG=∑rs∈EGdr+ds2.In 2012, Zhang introduced the harmonic index. It is defined as follows [16]:(7)HG=∑rs∈EG2dr+ds.After working on the Zagreb index, in 2015, Furtula and Gutman introduce the forgotten indexF(G):(8)FG=∑rs∈EGdr2+ds2.Ghorbani and Azimi proposed two new types of Zagreb indices of a graphG in 2012. PM1 is the first multiple Zagreb index, and PM2 is the second multiple Zagreb index [6]:(9)PM1G=∏rs∈EGdr+ds,(10)PM2G=∏rs∈EGdr×ds,M1G,p and M2G,p, the first Zagreb polynomial and the second Zagreb polynomial [12, 17], respectively, are defined as(11)M1G,p=∑rs∈EGpdr+ds,(12)M2G,p=∑rs∈EGpdr×ds.Recently, Furtula et al. proposed the second Zagreb index:(13)RM2G=∑rs∈EGdr−1ds−1.After the success of the ABC index, Furtula et al. put forward its modified version in 2010 that they somewhat inadequately named “augmented Zagreb index.” It is defined as follows [18]:(14)AZIG=∑rs∈EGdrdsdr+ds−23.The invariant RR seems to be first encountered in a paper by Favaron, Mah’eo. The reciprocal Randic index is defined as follows:(15)RRG=∑rs∈EGdrds.In the same manner, the reduced second Zagreb index (equation (13)), is related to the ordinary second Zagreb index (equation (12)). The reduced reciprocal Randic index might be viewed as the reduced analogue of the reciprocal Randic index (equation (15)):(16)RRRG=∑rs∈EGdr−1ds−1.Vukicevic et al. introduce the symmetric division deg index in 2010 [19, 20] as(17)SDDG=∑rs∈EGdr2+ds2drds.Vukicevic and Gašperov in [20] initiated the study on the inverse sum indeg index of a network. The inverse sum indeg index is defined as follows [21]:(18)SDDG=∑rs∈EGdrdsdr+ds.Another index which belongs to the 4th class of ABC index was invented by Ghorbani and Hosseinzadeh in 2010 [22] as(19)ABCG=∑rs∈EGSr+Ss−2SrSs.The fifth class of geometric-arithmetic index, denoted by GA5, was presented by Graovac et al. in 2011 [23] as(20)GAG=∑rs∈EG2SrSsSr+Ss.


## 2. Main Results of Boron B12 Graph

In this paper, we deal with the topological properties of boron B12. Boron B12 is a two-dimensional icosahedral network. A recent study of high-pressure solid boron affirmed that the icosahedral B12-containing structures are quite universal [8]. On the other hand, previous experimental and theoretical studies of freestanding boron clusters have shown that the B12 structure is unstable in the gas phase [9, 10].The molecular graph of Boron B12 is shown in Figure 1. There are 11mn+11m+11n+9 vertices and 24mn+22m+22n+18 edges.Figure 1
(n, m) unit of boron.Theorem 1.
LetG be the boron B12 network with m>1 and n>1, then(21)Mα=22m+2n2α+3mn+m+53α+3mn+2m+3n+43α+m+2n+14α+9mn+7m+6n+54α+9mn+7m+7n+35α.Proof.
InG, there are total 11mn+11m+11n+9 vertices. There are 2m+2n vertices of degree 2, 2mn+m+n+3 vertices of degree 3, 3mn+3m+3n+3 vertices of degree 4, and 36mn+5m+5n+3 vertices of degree 5. Since Mα is expressed in equation (2),(22)MαG=∑r∈VGdrα,(23)Mα=22m+2n2α+3mn+m+53α+3mn+2m+3n+43α+m+2n+14α+9mn+7m+6n+54α+9mn+7m+7n+35α.
In the following six theorems, we considered the Randic index, sum connectivity index, ABC index, GA index, harmonic index, reduced Randic index, forgotten index, symmetric division deg index, inverse sum indeg index, ABC4 index, and GA5 index. This is the edge partition of boron B12 on the basis of starting and ending vertices of each edge, and we proceed this edge division to compute the topological indices.Theorem 2.
LetG be the boron B12 network with m>1 and n>1, then(1)
RαG=2m+2n8α+2m+2n10α+3mn+m+512α+m+2n+116α+3mn+2m+3n+415α+9mn+7m20α+9mn+7m+7n+325α+6n+520α(2)
χαG=2m+2n6α+2m+2n7α+3mn+m+57α+m+2n+18α+3mn+2m+3n+48α+9mn+7m+6n+59α+9mn+7m+7n+310α(3)
ABCG=2m+2n2+1/63mn+m+515+1/53mn+2m+3n+410+1/4m+2n+16+1/109mn+7m35+1/106n+535+2/59mn+7m+7n+32(4)
GAG=2/32m+2n2+2/72m+2n10+4/73mn+m+53+1/43mn+2m+3n+415+4/99mn+7m+6n+55+9mn+8m+9n+4Proof.
The network of boron B12 has 24mn+22m+22n+18 number of edges. There are seven disjoint edge sets of edge set E(G) depending on the degrees of the end vertices, i.e., EG=E1G∪E2G∪E3G∪E4G∪E5G∪E6G∪E7G. The edge partition E1G holds 2m+2n edges rs, where dr=2 and ds=4, the edge partition E2G holds 2m+2n edges rs, where dr=2 and ds=5, the edge partition E3G holds 3mn+m+5 edges rs, where dr=3 and ds=4, the edge partition E4G holds 3mn+2m+3n+4 edges rs, where dr=3 and ds=5, the edge partition E5G holds m+2n+1 edges rs, where dr=ds=4, the edge partition E6G holds 9mn+7m+6n+5 edges rs, where dr=4 and ds=5, and the edge partition E7G holds 9mn+7m+7n+3 edges rs, where dr=ds=5. From formulas (1) and (3)–(5), we get the desired results.Theorem 3.
LetG be the boron B12 network with m>1 and n>1, then(1)
HMG=1678m+1676n+1968mn+1270(2)
PM1G=62m+2n×72m+2n×73mn+m+5×83mn+2m+3n+4×8m+2n+1×99mn+7m+6n+5×109mn+7m+7n+3(3)
PM2G=82m+2n×102m+2n×123mn+m+5×153mn+2m+3n+4×16m+2n+1×209mn+7m+6n+5×259mn+7m+7n+3Proof.
LetG be the network of boron B12. The edge set EG is distributed in seven categories which depend on the degrees of the end vertices. The first disjoint edge set E1G holds 2m+2n edges rs, where dr=2 and ds=4, the second disjoint edge set E2G holds 2m+2n edges rs, where dr=2 and ds=5, the third disjoint edge set E3G holds 3mn+m+5 edges rs, where dr=3 and ds=4, the forth disjoint edge set E4G holds 3mn+2m+3n+4 edges rs, where dr=3 and ds=5, the fifth disjoint edge set E5G holds m+2n+1 edges rs, where dr=ds=4, the sixth disjoint edge set E6G holds 9mn+7m+6n+5 edges rs, where dr=4 and ds=5, and the seventh disjoint edge set E7G holds 9mn+7m+7n+3 edges rs, where dr=ds=5. Now, E1G=e2,4, E2G=e2,5, E3G=e3,4, E4G=e3,5, E5G=e4,4, E6G=e4,5, and E7G=e5,5. We obtained results by using formulas (6), (9), and (10).Theorem 4.
LetG be the boron B12 network with m>1 and n>1, then(1)
M1G,p=2m+2np6+2m+2np7+3mn+m+5p7+m+2n+1p8+3mn+2mp8+3n+4p8+9mn+7m+6n+5p9+9mn+7m+7n+3p10(2)
M1G,p=2m+2np8+2m+2np10+3mn+m+5p12+3mn+2mp15+3n+4p15+m+2n+1p16+9mn+7m+6n+5p20+9mn+7m+7n+3p25Proof.
LetG be the network of boron B12. The edge set EG is distributed in seven categories which depend on the degree of end vertices of each edge. The disjoint set is represented by er,s. The first disjoint set is e2,4, the second disjoint set is e2,5, the third disjoint set is e3,4, the fourth disjoint set is e3,5, the fifth disjoint set is e4,4, the sixth disjoint set is e4,5, and the seventh disjoint set is e5,5. By using formulas (11) and (12), we obtained the required results.Theorem 5.
LetG be the boron B12 network with m>1 and n>1, then(1)
RM2G=241m+240n+294mn+179(2)
AZIG=12576690219/21952000mn+40043532353/84672000m+2230300883/4741632n+42526959271/118540800Proof.
LetG be the network of boron B12. The edge set EG is distributed in seven categories which depend on the degree of end vertices of each edge. The disjoint set is represented by er,s. The first disjoint set is e2,4, the second disjoint set is e2,5, the third disjoint set is e3,4, the fourth disjoint set is e3,5, the fifth disjoint set is e4,4, the sixth disjoint set is e4,5, and the seventh disjoint set is e5,5. We get the results by using formulas (13) and (14).Theorem 6.
LetG be the boron B12 network with m>1 and n>1, then(1)
HG=6589/1260m+731/140n+757/140mn+5531/1260(2)
RRG=22m+2n2+2m+2n10+23mn+m+53+3mn+2m15+3n+415+29mn+7m+6n+55+45mn+39m+43n+19(3)
RRRG=2m+2n3+23mn+2m2+23n+42+23mn+m+56+29mn+7m+6n+53+36mn+35m+38n+15Proof.
LetG be the network of boron B12. The edge set EG is distributed in seven categories which depend on the degree of end vertices of each edge. The disjoint set is represented by er,s. The first disjoint set is e2,4, the second disjoint set is e2,5, the third disjoint set is e3,4, the fourth disjoint set is e3,5, the fifth disjoint set is e4,4, the sixth disjoint set is e4,5, and the seventh disjoint set is e5,5. By using formulas (7), (15), and (16), we get the desired results.Theorem 7.
LetG be the boron B12 network with m>1 and n>1, then(1)
FG=860m+860n+996mn+648(2)
SDDG=1433/30m+479/10n+99/2mn+566/15(3)
ISIG=11603/252m+2575/25562n+2983/56mn+2311/63Proof.
LetG be the network of boron B12. The edge set EG is distributed in seven categories which depend on the degree of end vertices of each edge. The disjoint set is represented by er,s. The first disjoint set is e2,4, the second disjoint set is e2,5, the third disjoint set is e3,4, the fourth disjoint set is e3,5, the fifth disjoint set is e4,4, the sixth disjoint set is e4,5, and the seventh disjoint set is e5,5. We obtained the required results by using formulas (8), (17), and (18).
In the following two theorems, we considered the fourth atom-bond connectivity index and the fifth geometric-arithmetic index. This is the edge partition of boron B12 on the basis of the degree sum of neighbors of end vertices of each edge. We proceed this edge division to compute ABC4(G) and GA5(G).Theorem 8.
LetG be the boron B12 network with m>1 and n>1, then(24)ABC4G=1/22+4/2211547+4/273546−2/3+2/2222+1/2242+1/10m−110+1/114m+2n+11330+1/68m−1442+1/391m+314858+1/374m+113838+1/34m−1119+2/119m+1357+1/92m+2n+1851+1/12m+2n+115+1/68m+3527+2/115n−1345+4/2852n−2570+1/238m+16902+1/266m+38246+1/154m+32618+1/10m−114+1/8m+2n+111+1/68m−1782+3/190m+2n+1570+1/6mn30+1/35mn1190+1/6mn42+1/423mn−2n−146+1/92m+4n−3690+2/3n11+1/6m+33+1/506m+121758+1/846mn−m−2n−1602+1/23m+2n+346+1/202m−235+1/228m+2n+14674+2/4372n−24370+1/418m+316302+1/1382n−21794+1/1266mn−2m−2n1554+2/3n.Proof.
The network of boron B12 has 24mn+22m+22n+18 number of edges. There are 41 disjoint degree sum of neighbors of end vertices of each edge, i.e., e8,16, e8,17, e10,19, e10,20, e12,18, …, e24,24. By using formula (19), we get the desired result.Theorem 9.
LetG be the boron B12 network with m>1 and n>1, then(25)GA5G=1/20m+3391+1/19m+1357+1/39m+1374+1/19n−1345+1/172n−2285+2/29m+2n+1190+2/31m+1238+2/33m+3266+2/41m+3418+1/212n−2437+1/22m+2n+3483+2/45m+1506+2/15273+2/136mn−2m−2n42+6/37m+2n+138+6/19m−110+4/41m−1102+8/37m+2n+121+4/37m−185+8/39m+2n+123+8/33m+317+12/172n−22+2/3m−12+1/9m+377+2/3m+2n+12+4/25m−134+12/7mn3+1/2mn35+6/5mn6+6/412n−246+2/112m−230+4/43m+2n+1114+4/156mn−m−2n−114+4/23m+333+4/47m+4n−3138+8/1514+8/1922−n+3mn.Proof.
The network of boron B12 has 24mn+22m+22n+18 number of edges. There are 41 disjoint degree sum of neighbors of end vertices of each edge, i.e., e8,16, e8,17, e10,19, e10,20, e12,18, …, e24,24. By using formula (20), we get the desired result.


## 3. Conclusion

We have computed the following topological indices such as Zagreb index, the Randic index, sum connectivity index, ABC index, GA index, hyper-Zagreb index, multiple Zagreb indices, Zagreb polynomials, reduced second Zagreb index, augmented Zagreb index, harmonic index, reduced Randic index, reduced reciprocal Randic index, forgotten index,ABC4 index, and GA5 index for boron B12 structure.In cheminformatics, Randic index is used to study the organic compounds. It is correlated with physicochemical properties of alkane such as boiling points, surface area, and enthalpy of formation. ABC index is based on some physicochemical properties like the stability of cyclo-alkane as well as strain energy. GA index has more predictive power than Randic and ABC index. Multiple Zagreb indices and Zagreb polynomials are applied to predict the bioactivity of graphs. The forgotten index is correlated with some chemical properties relating to energies of molecular graphs. AZI index is a valuable predictive index which is used in the study of the heat of formation. RRR index is used to study the normal boiling points of the graphs.We have discussed the graph theoretically not experimentally. The results obtained in this paper provide a significant contribution to graph theory and correlate the chemical structure of boron B12 with the large amount of information about physicochemical properties.



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