On topological properties of sierpinski networks

Chaos, Solitons and Fractals 98 (2017) 199–204

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

On topological properties of sierpinski networks Muhammad Imran a,b,∗, Sabeel-e-Hafi b, Wei Gao c, Mohammad Reza Farahani d a

Department of Mathematical Sciences, United Arab Emirates University, P. O. Box 15551, Al Ain, United Arab Emirates Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan c School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China d Department of Applied Mathematics, Iran University of Science and Technology (IUST) Narmak, 16844, Tehran, Iran b

a r t i c l e

i n f o

Article history: Received 14 November 2016 Revised 19 February 2017 Accepted 16 March 2017

Keywords: Atom-bond connectivity index Geometric-arithmetic index Sierpinski network

a b s t r a c t Sierpinski graphs constitute an extensively studied class of graphs of fractal nature applicable in topology, mathematics of Tower of Hanoi, computer science, and elsewhere. A large number of properties like physico-chemical properties, thermodynamic properties, chemical activity, biological activity, etc. are determined by the chemical applications of graph theory. These properties can be characterized by certain graph invariants referred to as topological indices. In QRAR/QSPR study these graph invariants has played a vital role. In this paper, we study the molecular topological properties of Sierpinski networks and derive the analytical closed formulas for the atom-bond connectivity (ABC) index, geometric-arithmetic (GA) index, and fourth and fifth version of these topological indices for Sierpinski networks denoted by S(n, k). © 2017 Elsevier Ltd. All rights reserved.

1. Introduction and preliminary results The application of molecular structure descriptors is nowadays a standard procedure in the study of structure-property relations, especially in QSPR/QSAR study. In the last few years, the number of proposed molecular descriptors is rapidly growing due to the chemical significance of these descriptors. These descriptors correlate certain chemical and physical properties of chemical compounds. A close correlation of Randic´ index to the boiling point and Kovats constants has been found. A good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes is provided by the atom-bond connectivity (ABC) index. For certain physico-chemical properties like boiling point, Entropy, Enthalpy of vaporization, Standard enthalpy of vaporization, Enthalpy of formation and Acentric factor, the predictive power of geometric-arithmetic (GA) index is better than predictive power of the Randic´ connectivity index [8]. Topological characterization of chemical structures allows the classification of molecules and modelling unknown structures with desired properties. Molecules and molecular compounds are often modeled by molecular graphs. A model that is used to characterize a chemical compound is called chemical graph. A molecular graph is a representation of the structural formula of a chemical compound in

∗

Corresponding author. E-mail addresses: [email protected] (M. Imran), sabeel.hafi@yahoo.com (Sabeel-e-Hafi), [email protected] (W. Gao), [email protected] (M. Reza Farahani). http://dx.doi.org/10.1016/j.chaos.2017.03.036 0960-0779/© 2017 Elsevier Ltd. All rights reserved.

terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds. The combination of chemistry, mathematics and information science which studies QSAR/QSPR relationships is known as cheminformatics. Many chemical and physical properties of the chemical compounds can be found with the help of QSAR /QSPR models. The topological indices such as Wiener index, Szeged index, Randic´ index, Zagreb indices and ABC index are used to correlate different chemical and physical properties like the boiling point, molecular weight, vapour pressure, π -electrom energy etc. A molecular graph can be described by different ways such as, a drawing, a polynomial, a sequence of numbers, a matrix or by a derived number called a topological index . A topological index is a numeric quantity associated with a graph which characterizes the topology of graph and is invariant under the graph automorphism. To establish a correlation model between the structures of chemical compounds and the coressponding chemical and physical properties it is required to numerically code the structures of chemical compounds. hence in the QSAR/QSPR studied the task is to transfer the graph into numerical format. for this purpose there are many techniques in which the popular one is topological indices. In more precise way, a topological index Top(G) of a graph, is a number such that, if H is isomorphic to G, then T op(H ) = T op(G ). The concept of topological indices came from Wiener [30] while he was working on boiling point of paraffin, named this index as path number. Later on, the path number was renamed as Wiener index [7]. There are three major classes of topological indices which are distance based topological indices, degree based topological indices and counting

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related. Among these classes degree based topological indices are of great importance and play a vital role in chemical graph theory and particularly in theoretical chemistry. In this article, G is considered to be a graph with vertex set and edge set V(G) and E(G) respectively. The degree of a vertex u ∈ V(G) in a graph G is the number of edges which are incident with vertex u. It is denoted by du . The notations used in this article are mainly taken from the books [10,15]. Let G be a connected graph. Then the Wiener index [30] of G is defined as

W (G ) =

1  d (u, v ) 2

(1)

(u,v )

where (u, v) is any ordered pair of vertices in G and d(u, v) is the distance between the vertices u and v. Milan Randic´ [28] introduced the oldest degree based topological index Randi´c index, denoted by R− 1 (G ) and defined as 2

R− 1 ( G ) = 2





uv∈E ( G )

1

(2)

du dv

Randic´ index was then generalized by Bollobás and Erdös [5] for any real number α . The general Randic´ index, Rα (G), is defined as



Rα ( G ) =

(du dv )α for α ∈ R

(3)

uv∈E ( G )

Fig. 1. Sierpinski network S(3, 4)

et al. [29] and Ghorbani et al. [9] repectively, by using the eccentricity of each vertex of a connected graph. They are defined as





eu + ev − 2 eu ev

Gutman and Trinajstic´ [16] introduced the first and second Zagreb indices, denoted by M1 (G) and M2 (G), respectively and defined as

ABC5 (G ) =

M1 ( G ) =

Where eu = maxv∈V (G ) d (u, v ).



( du + dv )

(4)

uv∈E ( G )



M2 ( G ) =

GA4 (G ) =

(dG (u )dG (v )).

(5)

uv∈E ( G )

The general Randic´ index for α = 1 is the second Zagreb index for any graph. Estrada et al. [11] introduced a well known topological index, called the atom-bond connectivity (ABC) index. It is defined as



ABC (G ) =



uv∈E ( G )

du + dv − 2 du dv

(6)

The degree based topological index geometric-arithmetic (GA) index was introduced by Vukicˇ evic´ et al. in [29] and defined as



 2 du dv GA(G ) = du + dv

(7)

uv∈E ( G )

Some versions of the atom bond connectivity (ABC) index and geometric arithemetic GA index was defined recently. Ghorbani et al. [13] introduced the fourth version of ABC index and defined as

ABC4 (G ) =





uv∈E ( G )

where Su =



v∈NG (u )

Su + Sv − 2 , Su Sv

(8)

dv where NG (u ) = {v ∈ V (G ) | uv ∈ E (G )}.

The fifth version of GA index is proposed by Graovac et al. [14] and defined as

GA5 (G ) =

uv∈E ( G )

√  2 Su Sv Su + Sv

(9)

uv∈E ( G )

The eccentric version of atom-bond connectivity idex, ABC5 and geometric-arithmetic index, GA4 were introduced by Vukicˇ evic´

 2√eu ev eu + ev

uv∈E ( G )

Imran et al. studied various degree based topological indices for various networks like silicates, hexagonal, honeycomb and oxide in [20]. Nowadays there is an extensive research activity on ABC and GA indices and their variants. For further study of topological indices of various graphs and chemical structures, see [1–4,6,12,17– 19,21,22,24–27]. 2. On topological indices of sierpinski networks S(n, k) In this paper, we compute the analytical closed formulas for ABC, GA, ABC4 , ABC5 , GA4 and GA5 indices for Sierpinski networks denoted by S(n, k). The sierpinski networks S(n, k) are defined on the vertex set V (S(n, k )) = {1, 2, . . . ., k}n . We have |V (S(n, k ))| = kn for any n ≥ 1 and k ≥ 1 and vertices of these graphs can be written as u = (u1 , u2 , . . . .un ), where ur ∈ {1, 2, . . . . . . , k} and r ∈ {1, 2, . . . ..n}. Two different vertices u = (u1 , u2 , . . . .un ) and v= (v1 , v2 , . . . .vn ), where ur , vr ∈ {1, 2, . . . . . . , k} and r ∈ {1, 2, . . . ..n} of sierpinski network S(n, k) are adjacent iff there exists an h ∈ {1, 2, . . . ., n} such that: • ut = vt for t = 1, . . . , h − 1. • uh = vh ; and • ut = vh and vt = uh for t = h + 1, . . . , n. The definition of sierpinski networks S(n, k) originated from the topological studies of the Lipscomb’s space and sierpinski network S(n, k) is isomorphic to the graphs of the Tower of Hanoi with n disks [23]. Moreover, sierpinski graphs are the first nontrivial families of graphs of fractal type for which the crossing number is known and several metric invariants such as unique 1-perfect codes, average distance of sierpinski graphs are studied in the literature. The sierpinski networks S(3, 4) and S(2, 5) are depicted in

M. Imran et al. / Chaos, Solitons and Fractals 98 (2017) 199–204

201

Table 2 Edge partition of graph S(n, k) where n = 2 based on degree sum of vertices lying at unit distance from end vertices of each edge.

(k2 − k, k2 − 1 ) k2 − k

(Su , Sv ) where uv ∈ E(G) Number of edges

( k2 − 1, k2 − 1 ) k (kn −2k+1 ) 2

Now we compute the closed-form expression for the geometricarithmetic index (GA) of sierpinski network S(n, k). Theorem 2.0.2. Consider the graph of sierpinski network S(n, k); then its geometric-arithmetic index (GA) is as follows:

GA(S(n, k )) =

√ 4k k2 − k kn+1 − 5k + 2k − 1 2

Proof. The proof of this result is just calculation based. We prove it by using the Table 1. Since we have



 2 du dv GA(G ) = du + dv uv∈E ( G )

⇒ GA(S(n, k )) = (2k )

Table 1 Edge partition of S(n,k) Sierpinski network based on degrees of end vertices of each edge.

( k − 1, k ) 2k



(k, k) kn+1 −5k 2

( k − 1 )k

k−1+k



Fig. 2. Sierpinski network S(2, 5)

(du , dv ) where uv ∈ E(G) Number of edges

2

⇒ GA(S(n, k )) = (4k )

( k − 1 )k

2k − 1

+

 (kn+1 − 5k ) 2 (k )(k ) k+k

2 √

+

(kn+1 − 5k ) 2 k2 2

2k

After some more calculation we have the following closed formula

GA(S(n, k )) =

√ 4k k2 − k kn+1 − 5k + . 2k − 1 2

 Figs. 1 and 2. In the following theorem,an analytical closed formula for ABC index of sierpinski networks S(n, k) is derived. Theorem 2.0.1. The closed formula for the atom-bond connectivity (ABC) index of sierpinski network S(n, k) is as follows:



ABC (S(n, k )) = 2

k ( 2k − 3 ) + ( kn − 5 ) k−1



k−1 2

Proof. By using the edge partition based on the degrees of end vertices of each edge of graph of S(n, k) given in Table 1, we can compute the ABC index of sierpinski graph. Since we have

ABC (G ) =





uv∈E ( G )

du + dv − 2 du dv

By using Table 1, the proof is mechanical. Now we apply the formula for atom-bond connectivity index and do some calculation to get our result.



ABC (S(n, k )) = 2k

k−1+k−2 + ( k − 1 )k

 ⇒ ABC (S(n, k )) = 2k

2k − 3 + k2 − k





kn+1 − 5k 2

kn+1 − 5k 2





k+k−2 (k )(k )

2k − 2 k2

After some calculation, we have the following closed formula for atom-bond connectivity index for the sierpinski network. Hence we have the following closed expression of ABC index for sierpinski network.



ABC (S(n, k )) = 2 

k ( 2k − 3 ) + ( kn − 5 ) k−1



k−1 . 2

Next, we compute the fourth ABC index of sierpinski network S(n, k). The following theorem investigates the fourth ABC index for sierpinski network S(n, k). Theorem 2.0.3. Consider the graph of sierpinski network S(n, k); then its fourth ABC index is as follows:

⎧ ⎪ k ( 2k − 3 ) ⎪ √ ⎪ ⎪ k ( kn − 2k + 1 ) 2k2 − 4 ⎪ ⎪ + , ⎪ ⎪ 2 ( k2 − 1 ) ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎨ k ( k − 2 ) 2k2 − 4 k ( 2k − 3 ) + ABC4 (S(n, k )) = 2 (k + 1 )  ⎪ ⎪ ( k − 1 )( 2 k − 3) ⎪ ⎪ + ⎪ ⎪ ( k + 1 ) ⎪ ⎪ √ ⎪ ⎪ ( kn − k2 − k + 1 ) 2k2 − 2 ⎪ ⎪ + , ⎪ ⎪ 2k ⎩

n=2

n≥3

Proof. We prove it by using Tables 2 and 3. We use the edge partition of graph of S(n, k) based on the degree sum of vertices lying at unit distance from end vertices of each edge. Tables 2 and 3 explains such a partition for S(n, k) graphs for n = 2 and n ≥ 3 respectively. Now by using the partition given in Tables 2 and 3, we can apply the formula of ABC4 index to compute this index for sierpinski networks S(n, k). Since we have

ABC4 (G ) =

 uv∈E ( G )



Su + Sv − 2 Su Sv

 ⇒ ABC4 (S(n, k )) = (k − k ) 2

k2 − k + k2 − 1 − 2 (k2 − k )(k2 − 1 )

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M. Imran et al. / Chaos, Solitons and Fractals 98 (2017) 199–204 Table 3 Edge partition of graph S(n, k) where n ≥ 3 based on degree sum of vertices lying at unit distance from end vertices of each edge. (Su , Sv ) where uv ∈ E(G) Number of edges

+

k ( kn − 2k + 1 ) 2



 k ( kn − 2k + 1 )  2 2k − 4 2 ( k2 − 1 )

ABC4 (G ) =



uv∈E ( G )

Su + Sv − 2 Su Sv



+

k ( k2 − 3k + 2 ) 2



+ ( k2 − k )



k(kn − 2k + 1 ) 2 (k2 − 1 )(k2 − 1 ) + 2 ( k2 − 1 ) + ( k2 − 1 )



+

2k2 − k − 3 k(k − 1 )(k − 1 )(k + 1 )



+ (k − k ) 2

k (k − k − k + 1 ) 2



k ( kn − 2k + 1 ) 2 ( k2 − 1 )2 2 ( 2k2 − 2 )

After some calculation, we get the followings



2 ( k − 1 ) k3 ( k + 1 ) k ( kn − 2k + 1 ) GA5 (S(n, k )) = + . 2k + 1 2

2k2 − 3 k2 ( k2 − 1 )

2

Now for n ≥ 3, since we have

GA5 (G ) =

−2 k4

2k2

√  2 Su Sv Su + Sv

uv∈E ( G )

After some calculations, we have

√  k ( k − 2 ) 2k2 − 4 ABC4 (S(n, k )) = k(2k − 3 ) + 2 (k + 1 )  √ n 2 (k − 1 )(2k − 3 ) (k − k − k + 1 ) 2k2 − 2 + + (k + 1 ) 2k

⇒ GA5 (S(n, k )) = (k − k ) 2

+(

Next we compute the fifth version of GA index of sierpinski network S(n, k). The following theorem computes the fifth version of GA index of this family of sierpinski network. Theorem 2.0.4. Consider the graph of sierpinski network S(n, k); then its fifth version of GA index is as follows:



2



(k2 − k )(k2 − 1 )

k2 − k + k2 − 1

k(k − 3k + 2 )) 2 ( − 1 )(k2 − 1 ) ) 2 k2 − 1 + k2 − 1 2

+ (k − k ) 2



GA5 (S(n, k ))

k(k − 1 )(k − 1 )(k + 1 )) 2k2 − k − 1

⇒ GA5 (S(n, k )) = (2k(k − 1 ))

k ( k2 − k − 2k + 2 )  2 + 2k − 4 2 ( k2 − 1 )

+



(k2 − k )(k2 − 1 ) ( k2 − k ) + ( k2 − 1 )

2





n

n≥3

√  2 Su Sv Su + Sv

⇒ GA5 (S(n, k )) = (k − k )

k2 + k2 − 2 (k2 )(k2 )

⇒ ABC4 (S(n, k )) = (k2 − k )

2

3 ⎪ ⎪ √ ⎪ 2 ⎪ 2 k + 1 k ( kn − k2 − k + 1 ) 2 k ( k − 1 ) ⎪ ⎪ + + , ⎪ ⎪ 2 2k2 − 1 ⎩

2

k2 − 1 + k2 − 2 k2 ( k2 − 1 )



2k + 1

n=2

uv∈E ( G )

k2 − 1 + k2 − 1 − 2 (k2 − 1 )(k2 − 1 )

k ( kn − k2 − k + 1 ) + 2

k (kn −k2 −k+1 ) 2

 ⎧ 2 ( k − 1 ) k3 ( k + 1 ) k ( kn − 2k + 1 ) ⎪ ⎪ + , ⎪ ⎪ 2k+ 1 2 ⎪ ⎪ ⎪ 2(k − 1 ) k3 (k + 1 ) k(k − 1 )(k − 2 ) ⎪ ⎨ +

GA5 (G ) =

k2 − k + k2 − 1 − 2 (k2 − k )(k2 − 1 )

⇒ ABC4 (S(n, k )) = (k2 − k )



(k2 , k2 )

Proof. We prove it by using Tables 2 and 3. We use the edge partition of graph of S(n, k) based on the degree sum of vertices lying at unit distance from end vertices of each edge. Tables 2 and 3 explains such a partition for S(n, k) graphs for n = 2 and n ≥ 3 respectively. Now by using the partition given in Tables 2 and 3 we can apply the formula of GA5 index to compute this index for sierpinski networks S(n, k). For n = 2, since we have

√  k ( kn − 2k + 1 ) 2k2 − 4 ABC4 (S(n, k )) = k(2k − 3 ) + 2 ( k2 − 1 )



( k2 − 1, k2 ) k2 − k

k (k2 −3k+2 ) 2

=

2k2 − k − 3 k(k − 1 )(k − 1 )(k + 1 )

After some calculations, we have the following expression

and for n ≥ 3 Since we have

( k2 − 1, k2 − 1 )

k2 − 1 + k2 − 1 − 2 (k2 − 1 )(k2 − 1 )

⇒ ABC4 (S(n, k )) = (k(k − 1 )) +

(k2 − k, k2 − 1 ) k2 − k

2



k2

(k2 − 1 )(k2 )

k2 − 1 + k2



k(kn − k2 − k + 1 ) 2 (k2 )(k2 ) + 2 k2 + k2

After some calculations, we have



GA5 (S(n, k )) =

2(k − 1 ) k3 (k + 1 ) k(k − 1 )(k − 2 ) + 2k + 1 2 √ 3 2k2 ( k − 1 ) 2 k + 1 k ( kn − k2 − k + 1 ) + + . 2 2k2 − 1 

M. Imran et al. / Chaos, Solitons and Fractals 98 (2017) 199–204

In the following two theorems, we compute the analytical closed formula of the eccentric version of ABC and GA index for the sierpinski networks. Theorem 2.0.5. Consider the graph of sierpinski network S(n, k); then its eccentric version of ABC index denoted by ABC5 is as follows:

⎧ n+1 ⎪ ⎨k − k,

n=2

√

3

ABC5 (S(n, k )) =

(kn+1 − k ) 2n+1 − 4 ⎪ ⎩ , 2 ( 2n − 1 )

eu =

ABC5 (G ) =

 uv∈E ( G )

⇒ ABC5 (S(n, k )) =

kn+1 − k 2



3+3−2 (3 )(3 )

After some calculations, we have

kn+1 − k 3

ABC5 (S(n, k )) =

For every n ≥ 3, since we have

ABC5 (G ) =



 uv∈E ( G )

eu + ev − 2 eu ev

Where eu = maxv∈V (G ) d (u, v ).



⇒ ABC5 (S(n, k )) =

kn+1 − k 2



2n − 1 + 2n − 1 − 2 (2n − 1 )(2n − 1 )

After some calculations, we have

 In the next theorem, we have computed the closed formula for the GA4 index. Theorem 2.0.6. Consider the sierpinski network S(n, k); then the eccentric version of GA index denoted by GA4 is as follows:

kn+1 − k 2

Proof. The eccentricities of the vertices of S(n, k) are given as follows



eu =

n=2 n≥3

3 2n − 1,

For n = 2, since we have

GA4 (G ) =

 2√eu ev eu + ev

uv∈E ( G )

Where eu = maxv∈V (G ) d (u, v ).



⇒ GA4 (S(n, k )) =

kn+1 − k 2

  2

(3 )(3 ) 3+3

After some easy calculations, we have the following analytical closed formula

GA4 (S(n, k )) =

kn+1 − k 2

kn+1 − k 2(2n − 1 ) 2 2 ( 2n − 1 )

kn+1 − k 2

for all values of n and k.



Acknowledgements

√

(kn+1 − k ) 2n+1 − 4 ABC5 (S(n, k )) = 2 ( 2n − 1 )

GA4 (S(n, k )) =

⇒ GA4 (S(n, k )) =

The Randic´ index has been closely correlated with many chemical properties of the molecules and found to parallel the boiling point and Kovats constants. The atom-bond connectivity (ABC) index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. For certain physico-chemical properties, the predictive power of GA index is somewhat better than predictive power of the Randic´ connectivity index. In this paper, we have studied the molecular topological properties of sierpinski networks and derive the closed formulas for the atom-bond connectivity (ABC) index, geometric-arithmetic (GA) index, the fourth and fifth version of atom-bond connectivity index and the fourth fifth version of geometric-arithmetic index (GA5 ) for sierpinski networks denoted by S(n, k). We determined the analytical closed formula of ABC, ABC4 , ABC5 , GA, GA4 and GA5 indices for the sierpinsi networks S(n, k). Our main finding is that the relations found are quadratic and depends on the number of vertices in the network. Our main finding is that the relations found are in polynomials and depends on the number of vertices in the network.

eu + ev − 2 eu ev



  (kn+1 − k ) 2 (2n − 1 )(2n − 1 ) 2 ( 2n − 1 ) + ( 2n − 1 )

3. Conclusion

For n = 2, since we have





GA4 (S(n, k )) =

GA4 (S(n, k )) =

n=2 n≥3

3 2n − 1,

When n ≥ 3, for all values of k we have

Hence, we get the following expression

n≥3

Proof. The eccentricities of the vertices of S(n, k) are given as follows



203

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