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

Wei Gao; Muhammad Kamran Siddiqui; Muhammad Naeem; Muhammad Imran

doi:10.1515/phys-2018-0077

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

Wei Gao , Muhammad Kamran Siddiqui , Muhammad Naeem and Muhammad Imran

Published/Copyright: October 16, 2018

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From the journal Open Physics Volume 16 Issue 1

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)=∑uv∈E(G)du+dv−2dudv and GA(G)=∑uv∈E(G)2dudvdu+dv, respectively, where d_u 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 G_{p, 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 d_v.

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, v ∈ V(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)=∑uv∈E(G)Ju+Jv−2JuJv, where J_u 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)=∑uv∈E(G)du+dv−2dudv.(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)=∑uv∈E(G)Su+Sv−2SuSv.(2)

where Su=∑uv∈E(G)dv,Sv=∑uv∈E(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)=∑uv∈E(G)Mu+Mv−2MuMv.(3)

where Mu=∏uv∈E(G)dv,Mv=∏uv∈E(G)du.

A family Λ of geometric arithmetic topological indices consisting of elements(member) of the form GA(G)=∑uv∈E(G)2JuJvJu+Jv, where J_u 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)=∑uv∈E(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)=∑uv∈E(G)2SuSvSu+Sv.(5)

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

GAM(G)=∑uv∈E(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 G_p_,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 = K_n be the complete graph, then for all u ∈ V(K_n), the d_u = n − 1, so M_u = (n − 1)_n−1. Thus

ABCM(G)=n×(n−1)n−1−12(n−1)n−2GAM(G)=n(n−1)2.

Example 2

If G = C_n be the cycle graph, then for all u ∈ V(C_n), then d_u = 2, so M_u = 4. Thus

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

Example 3

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

ABCM(G)=n−546+42GAM(G)=n−1.

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.

$Figure 1 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.

(S_u, S_v)	Frequency
(4, 4)	4
(4, 5)	8
(5, 5)	2p + 2q – 8
(5, 7)	4p + 4q – 8
(7, 9)	2p + 2q – 4
(9, 9)	6pq − 7p − 7q + 8

Table 2

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

(M_u, M_v)	Frequency
(4, 4)	4
(4, 6)	8
(6, 6)	2p + 2q – 8
(6, 12)	4p + 4q – 8
(12, 27)	2p + 2q – 4
(27, 27)	6pq − 7p − 7q + 8

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 G≅Opq. Then the fourth atom bond connectivity index ABC₄(G) is given as

ABC4(G)=8pq3−28q9+4355+(2(2p+2q−8))25+(4p+4q−8)147+(2p+2q−4)23+329+6−28p9.

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)=∑uv∈E(G)Su+Sv−2SuSvABC4(Opq)=44+4−24×4+84+5−24×5+(2p+2q−8)5+5−25×5+(4p+4q−8)5+7−25×7+(2p+2q−4)7+9−27×9+(6pq−7p−7q+8)9+9−29×9.

After some easy calculations we get:

ABC4(Opq)=8pq3−28q9+4355+(2(2p+2q−8))25+(4p+4q−8)147+(2p+2q−4)23+329+6−28p9

Theorem 2

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

ABCM(Opq)=(6pq−7p−7q+8)5227+(2p+2q−8)106+(4p+4q−8)23+(2p+2q−4)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 ABC_M(G) is computed as:

ABCM(G)=∑uv∈E(G)Mu+Mv−2MuMvABCM(Opq)=44+4−24×4+84+6−24×6+(2p+2q−8)6+6−26×6+(4p+4q−8)6+12−26×12+(2p+2q−4)12+27−212×27+(6pq−7p−7q+8)27+27−227×27.

After some easy calculations we obtained:

ABCM(Opq)=(6pq−7p−7q+8)5227+(2p+2q−8)106+(4p+4q−8)23+(2p+2q−4)3718+6+833.

Theorem 3

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

GA5(Opq)=6pq−5p−5q+(8p+8q−16)3512+(4p+4q−8)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)=∑uv∈E(G)2SuSvSu+SvGA5(Opq)=(4)24×44+4+(8)24×54+5+(2p+2q−8)25×55+5+(4p+4q−8)25×75+7+(2p+2q−4)27×97+9+(6pq−7p−7q+8)29×99+9.

After some easy calculations we get:

GA5(Opq)=6pq−5p−5q+(8p+8q−16)3512+(4p+4q−8)6316+4+3259.

Theorem 4

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

GAM(Opq)=6pq−41p13−41q13+(8p+8q−16)23+413+1665.

Proof

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

GAM(G)=∑uv∈E(G)2MuMvMu+MvGAM(Opq)=(4)24×44+4+(8)24×64+6+(2p+2q−8)26×66+6+(4p+4q−8)26×126+12+(2p+2q−4)212×2712+27+(6pq−7p−7q+8)227×2727+27.

After some easy calculations we get:

GAM(Opq)=6pq−41p13−41q13+(8p+8q−16)23+413+1665.

4 The hexagonal grid H(p, q)

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.

S_u, S_v)	Frequency
(4, 5)	4
(5, 5)	q
(5, 7)	8
(5, 8)	2q − 4
(6, 7)	4p − 8
(7, 9)	2p
(8, 8)	q − 2
(8, 9)	2q − 4
(9, 9)	3pq − 4p − 4q + 5

Table 4

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

(M_u, M_v)	Frequency
(4, 6)	4
(6, 6)	q
(6, 12)	8
(6, 18)	2q − 4
(9, 12)	4p − 8
(12, 27)	2p
(18, 18)	q − 2
(18, 27)	2q − 4
(27, 27)	3pq − 4p − 4q + 5

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 G ≅ H(p, q). The fourth atom bond connectivity index ABC₄(G) of H(p, q) is given as

ABC4(H(p,q))=4pq3−16q9−16p9+(2q−4)11020+(4p−8)46242+22p3+(q−2)148+(2q−4)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)=∑uv∈E(G)Su+Sv−2SuSvABC4(H)=(4)4+5−24×5+(q)5+5−25×5+(8)5+7−25×7+(2q−4)5+8−25×8+(4p−8)6+7−26×7+(2p)7+9−27×9+(q−2)8+8−28×8+(2q−4)8+9−28×9+(3pq−4q−4p+5)9+9−29×9

One can easily calculate that

ABC4(H(p,q))=4pq3−16q9−16p9+(2q−4)11020+(4p−8)46242+22p3+(q−2)148+(2q−4)3012+2355+22q5+8147+209

Theorem 6

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

ABCM(H(p,q))=433+10q6+823+(2q−4)6618+(4p−8)5718+37p9+(q−2)3418+(2q−4)25854+(3pq−4q−4p+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+6−24×6+(q)6+6−26×6+(8)6+12−26×12+(2q−4)6+18−26×18+(4p−8)9+12−29×12+(2p)12+27−212×27+(q−2)18+18−218×18+(2q−4)18+27−218×27+(3pq−4q−4p+5)27+27−227×27

One can easily calculate that

ABCM(H(p,q))=433+10q6+823+(2q−4)6618+(4p−8)5718+37p9+(q−2)3418+(2q−4)25854+(3pq−4q−4p+5)5227

Theorem 7

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

GA5(H(p,q))=1659−2q+4353+(4q−8)4013+(8p−16)4213+3p74+3+(4q−8)7217+3pq−4p.

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+(2q−4)25×85+8+(4p−8)26×76+7+(2p)27×97+9+(q−2)28×88+8+(2q−4)28×98+9+(3pq−4p−4q+5)29×99+9.

After some easy calculations we get

GA5(H(p,q))=1659−2q+4353+(4q−8)4013+(8p−16)4213+3p74+3+(4q−8)7217+3pq−4p.

Theorem 8

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

GAM(H(p,q))=865−2q+1623+(4q−8)34+(8p−16)127−28p13+3+(4q−8)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+(2q−4)26×186+18+(4p−8)29×129+12+(2p)212×2712+27+(q−2)218×1818+18+(2q−4)218×2718+27+(3pq−4p−4q+5)227×2727+27.

After some computation, we obtained the following result

GAM(H(p,q))=865−2q+1623+(4q−8)34+(8p−16)127−28p13+3+(4q−8)65+3pq.

5 The square grid G_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 square grid G_p_,q and we shall compare the results obtained in the last section. For p, q ≥ 1 the square grid G_p_,q consists of p horizontal squares and q vertical squares, see Figure 3. One can easily see that in G_p_,q the number vertices and edges are pq + p + q + 1 and 2pq + p + q, respectively. In this paper the we consider G_p_,q for p, q _ 4. For this we shall use Table 5 and Table 6 . In Table 5 we have partitioned the edges of G_p_,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 G_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.

Figure 3

The square grid G_9,7.

5.1 Results for the square grid G_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 G_p_,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 G_p_,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 G_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 5

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

(S_u, S_v)	Frequency
(6, 9)	8
(9, 10)	8
(10, 10)	2p + 2q − 16
(9, 14)	8
(10, 15)	2p + 2q − 12
(14, 15)	8
(15, 15)	2p + 2q − 16
(15, 16)	2p + 2q − 12
(16, 16)	2pq − 7p − 7q + 24

Table 6

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

(M_u, M_v)	Frequency
(4, 6)	4
(6, 6)	q
(6, 12)	8
(6, 18)	2q − 4
(9, 12)	4p − 8
(12, 27)	2p
(18, 18)	q − 2
(18, 27)	2q − 4
(27, 27)	3pq − 4p − 4q + 5

Theorem 9

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

ABC4(Gp,q)=4789+417015+(2p+2q−16)1810+463+(2p+2q−12)13830+127035+(2p+2q−16)2815+(2p+2q−12)43560+(2pq−7p−7q+24)3016

Proof

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

ABC4(Gp,q)=(8)6+9−26×9+(8)9+10−29×10+(2p+2q−16)10+10−210×10+(8)9+14−29×14+(2p+2q−12)10+15−210×15+(8)14+15−214×15+(2p+2q−16)15+15−215×15+(2p+2q−12)15+16−215×16+(2pq−7q−7p+24)16+16−216×16

After simplification, we obtained required result:

ABC4(Gp,q)=4789+417015+(2p+2q−16)1810+463+(2p+2q−12)13830+127035+(2p+2q−16)2815+(2p+2q−12)43560+(2pq−7p−7q+24)3016

Theorem 10

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

ABCM(Gp,q)=21869+2879+(2p+2q−16)7036+2499+(2p+2q−12)678144+100236+(2p+2q−16)382192+(2p+2q−12)1338384+(2pq−7p−7q+24)510256

Proof

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

ABCM(Gp,q)=(8)9+24−29×24+(8)24+36−224×36+(2p+2q−16)36+36−236×36+(8)24+144−224×144+(2p+2q−12)36+192−236×192n+(8)144+192−2144×192n+(2p+2q−16)192+192−2192×192n+(2p+2q−12)192+256−2192×256n+(2pq−7q−7p+24)256+256−2256×256

After some calculation, we get:

ABCM(Gp,q)=21869+2879+(2p+2q−16)7036+2499+(2p+2q−12)678144+100236+(2p+2q−16)382192+(2p+2q−12)1338384+(2pq−7p−7q+24)510256

Theorem 11

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

GA5(Gp,q)=1665+481019−3p−3q−8+481423+(4p+4q−24)65+1621029+(4p+4q−24)12031+2pq.

Proof

Let G be the graph of G_p_,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+2q−16)210×1010+10+(8)29×149+14+(2p+2q−12)210×1510+15+(8)214×1514+15+(2p+2q−16)215×1515+15+(2p+2q−12)215×1615+16+(2pq−7p−7q+24)216×1616+16.

Simplification provide our required result as follows:

GA5(Gp,q)=1665+481019−3p−3q−8+481423+(4p+4q−24)65+1621029+(4p+4q−24)12031+2pq.

Theorem 12

For every p, q ≥ 4 consider the graph of G ≅ G_p_,q. Then the multiple geometric arithmetic index GA_M of G_p_,q is given as

GAM(Gp,q)=32326385−3p−3q−8+(66(4p+4q−24))3133+3237+2pq.

Proof

Let G be the graph of G_p_,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+2q−16)236×3636+36+(8)224×14424+144+(2p+2q−12)236×19236+192+(8)2144×192144+192+(2p+2q−16)2192×192192+192+(2p+2q−12)2192×256192+256+(2pq−7p−7q+24)2256×256256+256.

After some calculation we obtained our required result:

GAM(Gp,q)=32326385−3p−3q−8+(66(4p+4q−24))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 G_p_,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, ABC_M, GA₅ and GA_M for Opq, H(p, q) and G_p_,q, respectively.

Table 7

Comparison of ABC₄, ABC_M, GA₅ and GA_M for Opq.

[p, q]	ABC₄	ABC_M	GA₅	GA_M
[2,2]	15.123	13.260	27.808	15.073
[3,3]	30.657	24.764	59.664	58.308
[4,4]	51.526	39.474	103.52	101.54
[5,5]	77.728	57.388	159.38	156.78
[6,6]	109.26	78.506	227.23	224.01
[7,7]	146.13	102.83	307.09	303.24
[8,8]	188.33	130.36	398.95	394.49
[9,9]	235.88	161.09	502.81	497.72
[10,10]	288.74	195.04	618.66	612.95

Table 8

Comparison of ABC₄, ABC_M, GA₅ and GA_M for H(p, q).

[p, q]	ABC₄	ABC_M	GA₅	GA_M
[2,2]	7.4375	8.7535	18.832	18.154
[3,3]	19.201	15.326	37.747	36.650
[4,4]	30.963	23.5	62.661	61.147
[5,5]	45.392	33.276	93.578	91.644
[6,6]	62.487	44.656	130.50	128.14
[7,7]	82.250	57.638	173.41	170.64
[8,8]	104.68	72.221	222.34	219.14
[9,9]	129.78	88.408	277.25	273.64
[10,10]	157.54	106.20	338.15	334.13

Table 9

Comparison of ABC₄, ABC_M, GA₅ and GA_M for G_p,q.

[p, q]	ABC₄	ABC_M	GA₅	GA_M
[4,4]	16.494	8.8402	39.549	35.358
[5,5]	23.928	11.634	59.466	54.233
[6,6]	32.731	14.781	83.383	77.110
[7,7]	42.904	18.280	111.30	103.99
[8,8]	54.445	22.133	143.21	134.86
[9,9]	67.359	26.339	179.13	169.74
[10,10]	81.638	30.896	219.05	208.61

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.

References

[1] Siddiqui M.K., Imran M., Ahmad A., On zagreb indices, zagreb polynomials of some nanostar dendrimers, Appl. Math. Comput., 2016, 280, 132-139.10.1016/j.amc.2016.01.041Search in Google Scholar

[2] Wiener H., Influence of interatomic forces on para–n properties, J. Chem. Phys., 1947, 15, 766-767.10.1063/1.1746328Search in Google Scholar

[3] Randic M., On characterization of molecular branching, J. Amer. Chem. Soc., 1975, 97(23), 6609-6615.10.1021/ja00856a001Search in Google Scholar

[4] Furtula B., Graovac A., Vukicevic D., Atom–bond connectivity index of trees, Disc. Appl. Math., 2009, 157, 2828-2835.10.1016/j.dam.2009.03.004Search in Google Scholar

[5] Estrada E., Torres E., Rodriguez L., Gutman I., An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem., 1998, 37A, 849-855.Search in Google Scholar

[6] Ghorbani A., Hosseinzadeh M.A., Computing ABC₄ index of nanostar dendrimers, Optoelectr. Adv. Mat. Rapid Comm., 2010, 4, 1419-1422.Search in Google Scholar

[7] Vukičević D., Furtula B., Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem., 2009, 46, 1369-1376.10.1007/s10910-009-9520-xSearch in Google Scholar

[8] Ghorbani A., Ghorbani M., Hosseinzadeh M.A., Computing fifth geometric–arithmetic index for nanostar dendrimers, J. Math. Nanosci., 2011, 1, 33-42.Search in Google Scholar

[9] Ahmad M.S., Nazeer W., Kang S.M., Imran M., Gao W., Calculating degree-based topological indices of dominating David derived networks, Open Phys., 2017, 15, 1015-1021.10.1515/phys-2017-0126Search in Google Scholar

[10] Liu J.B., Siddiqui M.K., Zahid M.A., Naeem M., Baig A.Q., Topological Properties of Crystallographic Structure of Molecules, Symmetry, 2018, 10, 1-20.10.3390/sym10070265Search in Google Scholar

[11] Shao Z., Siddiqui M.K., Muhammad M.H., Computing Zagreb Indices and Zagreb Polynomials for Symmetrical Nanotubes, Symmetry, 2018, 10(7), 1-20.10.3390/sym10070244Search in Google Scholar

[12] Gao W., Siddiqui M.K., Imran M., Jamil M.K., Farahani M.R., Forgotten topological index of chemical structure in drugs, Saudi Pharmac. J., 2016, 24, 258-264.10.1016/j.jsps.2016.04.012Search in Google Scholar PubMed PubMed Central

[13] Gao W., Siddiqui M.K., Molecular Descriptors of Nanotube, Oxide, Silicate, and Triangulene Networks, J. Chem., 2017, 6540754, 1-10.10.1155/2017/6540754Search in Google Scholar

[14] Gao W., Siddiqui M.K., Naeem M., Rehman N.A., Topological Characterization of Carbon Graphite and Crystal Cubic Carbon Structures, Molecules, 2017, 22(9), 1496-1507.10.3390/molecules22091496Search in Google Scholar

[15] Rostray D.H., The modeling of chemical phenomena using topological indices, J. Comp. Chem., 1987, 8, 470-480.10.1002/jcc.540080427Search in Google Scholar

[16] Guariglia E., Entropy and Fractal Antennas, Entropy., 2016, 18(3), 84.10.3390/e18030084Search in Google Scholar

[17] Guariglia E., Spectral Analysis of the Weierstrass-Mandelbrot Function, IEEE Conference Proceedings, In: Proceeding of the 2^nd International Multidisciplinary Conference on Computer and Energy Science, Split, Croatia, 12-14, July, 2017.Search in Google Scholar

[18] Guariglia E., Fractional Derivative of the Riemann zeta function. In: Fractional Dynamics, Cattani C., Srivastava H., Yang X.J. (Eds.), De Gruyter, 2015, 357-368.10.1515/9783110472097-022Search in Google Scholar

[19] Shao Z., Wu P., Gao Y., Gutman I., Zhang X., On the maximum ABC index of graphs without pendent vertices, Appl. Math. Comput., 2017, 315, 298-312.10.1016/j.amc.2017.07.075Search in Google Scholar

[20] Shao Z., Wu P., Zhang X., Dimitrov, D., Liu, J., On the maximum ABC index of graphs with prescribed size and without pendent vertices, IEEE Access, 2018, 6, 27604-27616.10.1109/ACCESS.2018.2831910Search in Google Scholar

[21] Das K.C., Gutman I., Furtula B., Survey on geometric arithmetic indices of graphs, MATCH Commun. Math. Comput. Chem., 2011, 65, 595-644.Search in Google Scholar

[22] Gutman I., Ruscic B., Trinajstic N., Wilcox C.F., Graph theory and molecular orbitals, XII. Acyclic polyenes, J. Chem. Phys., 1975, 62, 3399-3405.10.1063/1.430994Search in Google Scholar

[23] Imran M., Siddiqui M.K., Naeem M., Iqbal M.A., On Topological Properties of Symmetric Chemical Structures, Symmetry., 2018, 10, 1-21.10.3390/sym10050173Search in Google Scholar

[24] Imran M., Ali M.A., Ahmad S., Siddiqui M.K., Baig A.Q., Topological Characterization of the Symmetrical Structure of Bismuth Tri-Iodide, Symmetry, 2018, 10(6), 1-20.10.3390/sym10060201Search in Google Scholar

[25] Gutman I., Trinajstc N., Graph theory and molecular orbitals., Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett.., 1972, 17, 535-538.10.1016/0009-2614(72)85099-1Search in Google Scholar

[26] Diudea M.V., Distance counting in tubes and tori: Wiener index and Hosoya polynomial, In: Nanostructures-NovelArchitecture, NOVA, New York, 2005, 203-242.Search in Google Scholar

[27] Stefu M., Diudea M.V., Wiener Index of C₄C₈ Nanotubes, MATCH Comm. Math. Comp. Chem., 2004, 50, 133-144.Search in Google Scholar

[28] Siddiqui M.K., Naeem M., Rahman N.A., Imran M., Computing topological indices of certain networks, J. Optoelctr. Adv. Material., 2016, 18(9), 884-892.Search in Google Scholar

Received: 2018-03-15

Accepted: 2018-07-15

Published Online: 2018-10-16

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Articles in the same Issue

https://doi.org/10.1515/phys-2018-0077

Keywords for this article

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

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

Article

Abstract

1 Introduction

Example 1

Example 2

Example 3

2 Applications of topological indices

3 The octagonal grid Opq

3.1 Results for the octagonal grid Opq

Theorem 1

Proof

Theorem 2

Proof

Theorem 3

Proof

Theorem 4

Proof

4 The hexagonal grid H(p, q)

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

Theorem 5

Proof

Theorem 6

Proof

Theorem 7

Proof

Theorem 8

Proof

5 The square grid Gp,q

5.1 Results for the square grid Gp,q

Theorem 9

Proof

Theorem 10

Proof

Theorem 11

Proof

Theorem 12

Proof

6 Comparison and conclusion

Acknowledgement

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

5 The square grid G_p,q

5.1 Results for the square grid G_p,q