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Open Physics

formerly Central European Journal of Physics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

Computing multiple ABC index and multiple GA index of some grid graphs

Wei Gao
  • Corresponding author
  • School of Information Science and Technology, Yunnan Normal University, Kunming, 650500, China
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  • Other articles by this author:
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/ Muhammad Kamran Siddiqui / Muhammad Naeem / Muhammad Imran
  • Department of Mathematical Sciences, United Arab Emirates University, Al Ain, P. O. Box 15551, United Arab Emirates
  • Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, 44000, Pakistan
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Published Online: 2018-10-16 | DOI: https://doi.org/10.1515/phys-2018-0077

Abstract

Topological indices are the atomic descriptors that portray the structures of chemical compounds and they help us to anticipate certain physico-compound properties like boiling point, enthalpy of vaporization and steadiness. The atom bond connectivity (ABC) index and geometric arithmetic (GA) index are topological indices which are defined as ABC(G)=uvE(G)du+dv2dudv and GA(G)=uvE(G)2dudvdu+dv, respectively, where du is the degree of the vertex u. The aim of this paper is to introduced the new versions of ABC index and GA index namely multiple atom bond connectivity (ABC) index and multiple geometric arithmetic (GA) index. As an application, we have computed these newly defined indices for the octagonal grid Opq, the hexagonal grid H(p, q) and the square grid Gp, q. Also, we compared these results obtained with the ones by other indices like the ABC4 index and the GA5 index.

Keywords: Atom-bond-connectivity index; geometric-arithmetic index; octagonal grid; hexagonal grid; square grid

PACS: 02.10.Ox

1 Introduction

There are sure concoction exacerbates that are helpful for the survival of living things. Carbon, oxygen, hydrogen and nitrogen are the primary components that aides in the generation of cells in the living things. Carbon is a fundamental component for human life. It is helpful in the arrangements of proteins, sugars and nucleic acids. It is crucial for the development of plants as carbon dioxide. The carbon atoms can bond together in different ways, called allotropes of carbon. The outstanding structures are graphite and jewel. As of late, numerous new structures have been found including nanotubes, buckminster fullerene and sheets, precious stone cubic structure, and so forth. The utilizations of various allotropes of carbon are talked about in detail in [1].

A graph G is simply a collection of points and lines that connect the points or subset of points. The points are called vertices of G and lines are called edges of G. The vertices set and edges set of G are denoted as V(G) and E(G), respectively. If e is an edge of G that connects the vertices u and v, then we can write e = uv. A graph is called connected graph if there is a path between all pairs of vertices. The degree of a vertex v in the graph G is the number of edges which are incident to the vertex v and will be represented by dv.

Let Γ be the family of finite graphs. A function T from Γ into set of real numbers having T(G) = T(H) property, for isomorphic G and H, is called a topological index. Someone can clearly notice that the vertices cardinality and the edges cardinality are topological indices. The earliest known topological index is Wiener index [2] and its based on distance, it is characterized as the sum of the half of distances between every pairs of vertices in a graph.

If u, vV(G), then the distance between the vertices u and v is given by the length of any arbitrary shortest path in G that connects u and v. Another well known and one of the earliest degree dependent index was due to Milan Randi´c [3] in 1975, characterized as the sum of the negative square root of the product of degree of the end vertices of each edge of the graph.

One can define the family of atom bond connectivity topological indices [4] consisting of elements(member) of the form ABC(G)=uvE(G)Ju+Jv2JuJv, where Ju is some number that in a uniquely way can be assigned with the vertex u of graph G. One of the element of Γ is the atom bond connectivity index introduced by Estrada et al. [5]:

ABC(G)=uvE(G)du+dv2dudv.(1)

Another well known member of Γ is the fourth version of atom bond connectivity denoted as ABC4 topological index of a graph G, introduced by Ghorbhani et.al. [6]:

ABC4(G)=uvE(G)Su+Sv2SuSv.(2)

where Su=uvE(G)dv,Sv=uvE(G)du.

Here, we define a new member of this family Γ, namely multiple atom bond connectivity index and it is defined as follows:

ABCM(G)=uvE(G)Mu+Mv2MuMv.(3)

where Mu=uvE(G)dv,Mv=uvE(G)du.

A family Λ of geometric arithmetic topological indices consisting of elements(member) of the form GA(G)=uvE(G)2JuJvJu+Jv, where Ju is some number that in a uniquely way can be assigned with the vertex u of G. One of the other member of Λ is the geometric arithmetic index GA of a graph G introduced by Vukičević et.al. [7]:

GA(G)=uvE(G)2dudvdu+dv.(4)

Another well known member of Λ is the fifth version of geometric arithmetic index and is denoted by GA5 topological index of a graph G, introduced by Graovoc et.al. [8]:

GA5(G)=uvE(G)2SuSvSu+Sv.(5)

Here, we define a new member of Λ namely multiple geometric -arithmetic index and it is characterized as:

GAM(G)=uvE(G)2MuMvMu+Mv.(6)

For more information and properties of topological indices, see [9 10 11 12 13 14 15].

Moreover this idea of computing the topological indices is helpful to discuss the concept of entropy. The entropy of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied [16]. Fractional derivative of the Riemann zeta function has been explicitly computed and the convergence of the real part and imaginary parts are studied with the help of topological indices [17, 18].

The aim of this paper is the introduction of the multiple atom bond connectivity index and multiple geometric arithmetic index. As an application we shall compute these new indices for the octagonal grid Opq, the hexagonal grid H(p, q) and the square grid Gp,q. Also, we compared these results obtained with the ones obtained by other indices like fourth atom bond connectivity index and fifth geometric arithmetic index via their computation too. But first we shall see some examples.

Example 1

Let G = Kn be the complete graph, then for all uV(Kn), the du = n − 1, so Mu = (n − 1)n−1. Thus

ABCM(G)=n×(n1)n112(n1)n2GAM(G)=n(n1)2.

Example 2

If G = Cn be the cycle graph, then for all uV(Cn), then du = 2, so Mu = 4. Thus

ABCM(G)=n46GAM(G)=n.

Example 3

If G = Pn, n ≥ 5 be the path graph of length n, then for all uV(Pn), we can compute easily as:

ABCM(G)=n546+42GAM(G)=n1.

2 Applications of topological indices

The atom bond connectivity index (ABC) is a topological descriptor that has correlated with a lot of chemical characteristics of the molecules and has been found to the parallel to computing the boiling point and Kovats constants of the molecules. Moreover, the atom bond connectivity (ABC) index provides a very good correlation for the stability of linear alkanes as well as the branched alkanes and for computing the strain energy of cyclo alkanes [19, 20]. To correlate with certain physico-chemical properties, GA index has much better predictive power than the predictive power of the Randic connectivity index [21]. The first Zagreb index and second Zagreb index were found to occur for computation of the total π-electron energy of the molecules within specific approximate expressions [22 23 24]. These are among the graph invariants, who were proposed for measurement of skeleton of branching of the carbon-atom [25].

3 The octagonal grid Opq

In [26] and [27] Diudea et. al. constitute a C4C8 net consisting of a trivalent decoration constructed by alternating octagons and squares in two different manners. One is by alternating squares C4 and octagons C8 in different ways denoted by C4C8(S) and other is by alternating rhombus and octagons in different ways denoted by C4C8(R).We denote C4C8(R) by Opq see Figure 1. In [28] they also called it as the Octagonal grid.

The octagonal grid O85 $O_{8}^5$
Figure 1

The octagonal grid O85

For p, q ≥ 1 the octagonal grid Opq, is the grid with p horizontal octagons and q vertical octagons. Therefore, in Opq the number vertices and edges are 4pq + 2p + 2q and 6pq + p + q, respectively. In this paper, we consider Opq for p, q ≥ 2.

Table 1

Partition of edges Opq based on sum of degrees belonging to neighbourhood of each vertex.

Table 2

Partition of edges Opq based on product of degree belonging to neighbourhood of each vertex.

3.1 Results for the octagonal grid Opq

Now we shall compute fourth atom bond connectivity index, fifth geometric arithmetic index, multiple atom bond connectivity index and multiple geometric arithmetic index and we shall compare the results obtained for Opq with for p, q ≥ 2. For this we shall use Table 1 and Table 2. In Table 1 we have partitioned the edges of based on the sum of degrees for each pair of vertices incident to same edge. In Table 2 we have partitioned the edges of Opq based on product of degrees of the neighbouring vertices to each pair of vertices incident to same edge. This will help us to develop the theorems of present section.

Theorem 1

For every p, q ≥ 2, consider the graph of GOpq. Then the fourth atom bond connectivity index ABC4(G) is given as

ABC4(G)=8pq328q9+4355+(2(2p+2q8))25+(4p+4q8)147+(2p+2q4)23+329+628p9.

Proof

Let G be the graph of Opq. Then by using Table 1 and equation (2), the fourth atom bond connectivity index ABC4(G) is computed below.

ABC4(G)=uvE(G)Su+Sv2SuSvABC4(Opq)=44+424×4+84+524×5+(2p+2q8)5+525×5+(4p+4q8)5+725×7+(2p+2q4)7+927×9+(6pq7p7q+8)9+929×9.

After some easy calculations we get:

ABC4(Opq)=8pq328q9+4355+(2(2p+2q8))25+(4p+4q8)147+(2p+2q4)23+329+628p9

Theorem 2

If GOpq for every p, q ≥ 2, then the multiple atom bond connectivity index ABCM(G) is :

ABCM(Opq)=(6pq7p7q+8)5227+(2p+2q8)106+(4p+4q8)23+(2p+2q4)3718+6+833.

Proof

Let G be the graph of Opq. Then by using Table 2 and equation (3), the multiple atom bond connectivity index ABCM(G) is computed as:

ABCM(G)=uvE(G)Mu+Mv2MuMvABCM(Opq)=44+424×4+84+624×6+(2p+2q8)6+626×6+(4p+4q8)6+1226×12+(2p+2q4)12+27212×27+(6pq7p7q+8)27+27227×27.

After some easy calculations we obtained:

ABCM(Opq)=(6pq7p7q+8)5227+(2p+2q8)106+(4p+4q8)23+(2p+2q4)3718+6+833.

Theorem 3

If GOpq for every p, q ≥ 2, then the fifth geometric arithmetic index GA5(G) is:

GA5(Opq)=6pq5p5q+(8p+8q16)3512+(4p+4q8)6316+4+3259.

Proof

Let G be the graph of Opq. Then by using Table 1 and equation (5), the fifth geometric arithmetic index GA5(G) is computed as below:

GA5(G)=uvE(G)2SuSvSu+SvGA5(Opq)=(4)24×44+4+(8)24×54+5+(2p+2q8)25×55+5+(4p+4q8)25×75+7+(2p+2q4)27×97+9+(6pq7p7q+8)29×99+9.

After some easy calculations we get:

GA5(Opq)=6pq5p5q+(8p+8q16)3512+(4p+4q8)6316+4+3259.

Theorem 4

For every p, q ≥ 2 consider the graph of GOpq. The multiple geometric arithmetic index GAM(G) is:

GAM(Opq)=6pq41p1341q13+(8p+8q16)23+413+1665.

Proof

Let G be the graph of Opq. Then by using Table 2 and equation (6). the multiple geometric arithmetic index GAM(G) is computed below:

GAM(G)=uvE(G)2MuMvMu+MvGAM(Opq)=(4)24×44+4+(8)24×64+6+(2p+2q8)26×66+6+(4p+4q8)26×126+12+(2p+2q4)212×2712+27+(6pq7p7q+8)227×2727+27.

After some easy calculations we get:

GAM(Opq)=6pq41p1341q13+(8p+8q16)23+413+1665.

4 The hexagonal grid H(p, q)

In this section we shall compute fourth atom bond connectivity index, fifth geometric arithmetic index, multiple atom bond connectivity index and multiple geometric arithmetic index for hexagonal grid H(p, q). Also we shall compare these results obtained in the last section. For p, q ≥ 1 the hexagonal grid H(p, q) consists of p octagons in a row (horizontal) and q represents the number of rows see Figure 2. One can easily see that in H(p, q) the number vertices and edges are 2pq + 2p + 2q and 3pq+2p+2q−1, respectively. In this paper the we consider H(p, q) for p, q ≥ 2.

The hexagonal grid H(10, 6).
Figure 2

The hexagonal grid H(10, 6).

For this we shall use Table 3 and Table 4. In Table 3 we have partitioned the edges of H(p, q) based on the sum of degrees for each pair of vertices incident to same edge. In Table 4 we have partitioned the edges of H(p, q) based on product of degrees of the neighbouring vertices to each pair of vertices incident to same edge. This will help us to develop the theorems of present section.

Table 3

Partition of edges H(p, q) based on sum of degrees belonging to neighbourhood of each vertex.

Table 4

Partition of edges H(p, q) based on product of degree belonging to neighbourhood of each vertex.

4.1 Results for the hexagonal grid H(p, q)

Now we shall compute fourth atom bond connectivity index, fifth geometric arithmetic index, multiple atom bond connectivity index and multiple geometric arithmetic index and we shall compare the results obtained for H(p, q) with for p, q ≥ 2. For this we shall use Table 3 and Table 4. In Table 3 we have partitioned the edges of H(p, q) based on the sum of degrees for each pair of vertices incident to same edge. In Table 4 we have partitioned the edges of H(p, q) based on product of degrees of the neighbouring vertices to each pair of vertices incident to same edge. This will help us to develop the theorems of present section.

Theorem 5

For every p, q ≥ 2 consider the graph of GH(p, q). The fourth atom bond connectivity index ABC4(G) of H(p, q) is given as

ABC4(H(p,q))=4pq316q916p9+(2q4)11020+(4p8)46242+22p3+(q2)148+(2q4)3012+2355+22q5+8147+209

Proof

Let G be the graph of H(p, q). Then as in Theorem 1, by using Table 3, equation (1) and following computations, the result follows:

ABC4(G)=uvE(G)Su+Sv2SuSvABC4(H)=(4)4+524×5+(q)5+525×5+(8)5+725×7+(2q4)5+825×8+(4p8)6+726×7+(2p)7+927×9+(q2)8+828×8+(2q4)8+928×9+(3pq4q4p+5)9+929×9

One can easily calculate that

ABC4(H(p,q))=4pq316q916p9+(2q4)11020+(4p8)46242+22p3+(q2)148+(2q4)3012+2355+22q5+8147+209

Theorem 6

For every p, q ≥ 2 consider the graph of GH(p, q). The multiple atom bond connectivity index ABCM(G) of H(p, q) is given as

ABCM(H(p,q))=433+10q6+823+(2q4)6618+(4p8)5718+37p9+(q2)3418+(2q4)25854+(3pq4q4p+5)5227

Proof

Let G be the graph of H(p, q). Then as in Theorem 2, by using Table 4, equation (2) and following computations, the result follows.

ABCM(H(p,q))=(4)4+624×6+(q)6+626×6+(8)6+1226×12+(2q4)6+1826×18+(4p8)9+1229×12+(2p)12+27212×27+(q2)18+18218×18+(2q4)18+27218×27+(3pq4q4p+5)27+27227×27

One can easily calculate that

ABCM(H(p,q))=433+10q6+823+(2q4)6618+(4p8)5718+37p9+(q2)3418+(2q4)25854+(3pq4q4p+5)5227

Theorem 7

For every p, q ≥ 2 consider the graph of GH(p, q). The fifth geometric arithmetic index GA5 of H(p, q) is given as

GA5(H(p,q))=16592q+4353+(4q8)4013+(8p16)4213+3p74+3+(4q8)7217+3pq4p.

Proof

Let G be the graph of H(p, q). Then as in Theorem 3, by using Table 3, equation (3) and the following computations, the result follows.

GA5(H(p,q))=(4)24×54+5+(q)25×55+5+(8)25×75+7+(2q4)25×85+8+(4p8)26×76+7+(2p)27×97+9+(q2)28×88+8+(2q4)28×98+9+(3pq4p4q+5)29×99+9.

After some easy calculations we get

GA5(H(p,q))=16592q+4353+(4q8)4013+(8p16)4213+3p74+3+(4q8)7217+3pq4p.

Theorem 8

For every p, q ≥ 2 consider the graph of GH(p, q). The multiple geometric arithmetic index GAM of H(p, q) is given as

GAM(H(p,q))=8652q+1623+(4q8)34+(8p16)12728p13+3+(4q8)65+3pq.

Proof

Let G be the graph of H(p, q). Then as in Theorem4, by using Table 4, equation (4) and the calculations below, the result follows.

GAM(H(p,q))=(4)24×64+6+(q)26×66+6+(8)26×126+12+(2q4)26×186+18+(4p8)29×129+12+(2p)212×2712+27+(q2)218×1818+18+(2q4)218×2718+27+(3pq4p4q+5)227×2727+27.

After some computation, we obtained the following result

GAM(H(p,q))=8652q+1623+(4q8)34+(8p16)12728p13+3+(4q8)65+3pq.

5 The square grid Gp,q

In this section we shall compute fourth atom bond connectivity index, fifth geometric arithmetic index, multiple atom bond connectivity index and multiple geometric arithmetic index for square grid Gp,q and we shall compare the results obtained in the last section. For p, q ≥ 1 the square grid Gp,q consists of p horizontal squares and q vertical squares, see Figure 3. One can easily see that in Gp,q the number vertices and edges are pq + p + q + 1 and 2pq + p + q, respectively. In this paper the we consider Gp,q for p, q _ 4. For this we shall use Table 5 and Table 6 . In Table 5 we have partitioned the edges of Gp,q based on the sum of degrees for each pair of vertices incident to same edge. In Table 6 we have partitioned the edges of Gp,q based on product of degrees of the neighbouring vertices to each pair of vertices incident to same edge. This will help us to develop the theorems of present section.

The square grid G9,7.
Figure 3

The square grid G9,7.

5.1 Results for the square grid Gp,q

Now we shall compute fourth atom bond connectivity index, fifth geometric arithmetic index, multiple atom bond connectivity index and multiple geometric arithmetic index and we shall compare the results obtained for Gp,q with for p, q ≥ 4. For this we shall use Table 5 and Table 6. In Table 5 we have partitioned the edges of Gp,q based on the sum of degrees for each pair of vertices incident to same edge. In Table 6 we have partitioned the edges of Gp,q based on product of degrees of the neighbouring vertices to each pair of vertices incident to same edge. This will help us to develop the theorems of present section.

Table 5

Partition of edges Gp,q based on sum of degrees belonging to neighbourhood of each vertex.

Table 6

Partition of edges Gp,q based on product of degree belonging to neighbourhood of each vertex.

Theorem 9

For every p, q ≥ 4 consider the graph of GGp,q. Then the fourth atom bond connectivity index ABC4(G) of Gp,q is given as

ABC4(Gp,q)=4789+417015+(2p+2q16)1810+463+(2p+2q12)13830+127035+(2p+2q16)2815+(2p+2q12)43560+(2pq7p7q+24)3016

Proof

Let G be the graph of Gp,q. Then as in Theorem 1, by using Table 5, equation (1) and the computations below, the result follows.

ABC4(Gp,q)=(8)6+926×9+(8)9+1029×10+(2p+2q16)10+10210×10+(8)9+1429×14+(2p+2q12)10+15210×15+(8)14+15214×15+(2p+2q16)15+15215×15+(2p+2q12)15+16215×16+(2pq7q7p+24)16+16216×16

After simplification, we obtained required result:

ABC4(Gp,q)=4789+417015+(2p+2q16)1810+463+(2p+2q12)13830+127035+(2p+2q16)2815+(2p+2q12)43560+(2pq7p7q+24)3016

Theorem 10

For every p, q ≥ 4 consider the graph of G ≅= Gp,q. Then the multiple atom bond connectivity index ABCM(G) of Gp,q is given as

ABCM(Gp,q)=21869+2879+(2p+2q16)7036+2499+(2p+2q12)678144+100236+(2p+2q16)382192+(2p+2q12)1338384+(2pq7p7q+24)510256

Proof

Let G be the graph of Gp,q. Then as in Theorem 2, by using Table 6, equation (2) and the computations below, the result follows.

ABCM(Gp,q)=(8)9+2429×24+(8)24+36224×36+(2p+2q16)36+36236×36+(8)24+144224×144+(2p+2q12)36+192236×192n+(8)144+1922144×192n+(2p+2q16)192+1922192×192n+(2p+2q12)192+2562192×256n+(2pq7q7p+24)256+2562256×256

After some calculation, we get:

ABCM(Gp,q)=21869+2879+(2p+2q16)7036+2499+(2p+2q12)678144+100236+(2p+2q16)382192+(2p+2q12)1338384+(2pq7p7q+24)510256

Theorem 11

For every p, q ≥ 4 consider the graph of GGp,q. Then the fifth geometric arithmetic index GA5 of Gp,q is given as

GA5(Gp,q)=1665+4810193p3q8+481423+(4p+4q24)65+1621029+(4p+4q24)12031+2pq.

Proof

Let G be the graph of Gp,q. Then as in Theorem 3, by using Table 5, equation (3) and the computations below, the result follows.

GA5(Gp,q)=(8)26×96+9+(8)29×109+10+(2p+2q16)210×1010+10+(8)29×149+14+(2p+2q12)210×1510+15+(8)214×1514+15+(2p+2q16)215×1515+15+(2p+2q12)215×1615+16+(2pq7p7q+24)216×1616+16.

Simplification provide our required result as follows:

GA5(Gp,q)=1665+4810193p3q8+481423+(4p+4q24)65+1621029+(4p+4q24)12031+2pq.

Theorem 12

For every p, q ≥ 4 consider the graph of GGp,q. Then the multiple geometric arithmetic index GAM of Gp,q is given as

GAM(Gp,q)=323263853p3q8+(66(4p+4q24))3133+3237+2pq.

Proof

Let G be the graph of Gp,q. Then as in Theorem 4, by using Table 6, equation (4) and the computations below, the result follows.

GAM(Gp,q)=(8)29×249+24+(8)224×3624+36+(2p+2q16)236×3636+36+(8)224×14424+144+(2p+2q12)236×19236+192+(8)2144×192144+192+(2p+2q16)2192×192192+192+(2p+2q12)2192×256192+256+(2pq7p7q+24)2256×256256+256.

After some calculation we obtained our required result:

GAM(Gp,q)=323263853p3q8+(66(4p+4q24))3133+3237+2pq.

6 Comparison and conclusion

In this paper, we have introduced the multiple atom bond connectivity index and multiple geometric arithmetic index. As their application we have computed these new indices for octagonal grid Opq hexagonal grid H(p, q) and square grid Gp,q. We then give comparisons of the results obtained by these indices with the ones obtained by other indices like the fourth atom bond connectivity index and the fifth geometric arithmetic index via their computation too for the octagonal grid, hexagonal grid and square grid graphs. Table 7, Table 8 and Table 9 show the comparisons between ABC4, ABCM, GA5 and GAM for Opq, H(p, q) and Gp,q, respectively.

Table 7

Comparison of ABC4, ABCM, GA5 and GAM for Opq.

Table 8

Comparison of ABC4, ABCM, GA5 and GAM for H(p, q).

Table 9

Comparison of ABC4, ABCM, GA5 and GAM for Gp,q.

Acknowledgement

The authors are grateful to the anonymous referees for their valuable comments and suggestions that improved this paper. This research is supported by the Start-Up Research Grant 2016 of United Arab Emirates University (UAEU), Al Ain, United Arab Emirates via Grant No. G00002233 and UPAR Grant of UAEU via Grant No. G00002590.

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About the article

Received: 2018-03-15

Accepted: 2018-07-15

Published Online: 2018-10-16


Citation Information: Open Physics, Volume 16, Issue 1, Pages 588–598, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0077.

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© 2018 Wei Gao et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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