Multiplicative Connectivity Banhatti Indices of Dendrimer Nanostars

Multiplicative Connectivity Banhatti Indices of

Dendrimer Nanostars

V. R. Kulli

Department of Mathematics,

Gulbarga University, Gulbarga 585106, INDIA. email: vrkulli @gmail.com

(Received on: February 26, 2018)

ABSTRACT

In Chemical Science, the multiplicative connectivity indices are used in the analysis of drug molecular structures which are helpful to find out the biological and chemical characteristics of drugs. In this paper, we compute the multiplicative product connectivity Banhatti index, multiplicative sum connectivity Banhatti index, multiplicative atom bond connectivity Banhatti index and multiplicative geometric-arithmetic index of certain infinite classes of dendrimer nanostars.

Mathematics Subject Classification : 05C05, 05C07, 05C35.

Keywords: molecular graph, multiplicative connectivity Banhatti indices, dendrimer nanostars.

1. INTRODUCTION

Let G be a finite, simple connected graphwith vertex set V(G) and edge set E(G). The degree dG(v) of a vertex v is the number of vertices adjacent to v. We refer to1 _{for undefined}

term and notation.

Motivated by the definition of the multiplicative atom bond connectivity index3 _and

its wide applications, Kulli4 _{introduced following multiplicative connectivity Banhatti indices}

of a molecular graph:

The multiplicative product connectivity Banhatti index of a graph G is defined as

 

   

1 .

ue G G

PBII G

d u d e





The multiplicative sum connectivity Banhatti index of a graph G is defined as

 

 

 

1

.

ue _{G G}

SBII G

d u d e







The multiplicative atom bond connectivity Banhatti index of a graph G is defined as

 

 

_{   }

 

G G

G G

ue

d u d e

ABCBII G

d u d e

 





The multiplicative geometric-arithmetic Banhatti index of a graph G is defined as

 

_{ }

   

_{ }

G G

ue

d u d e

GABII G

d u d e







Recently, many K Banhatti indices were studied, for example, in5,6,7,8,9,10,11,12,13_{. Also}

some multiplicative connectivity indices were studied, for example, in14,15,16,17,18,19,20,21,22_.

In this paper, the multiplicative connectivity Banhatti indices for certain infinite families of dendrimer nanostars are determined. For more information about dendrimer nanostars see23_.

2. RESULTS FOR DENDRIMER NANOSTARS D1[n]

In this section, we consider a family of dendrimer nanostars with n growth stages, denoted by D1[n], where n0. The molecular graph of D1[n] with 4 growth stages is depicted

in Figure 1.

Let G be the molecular graph of dendrimer nanostar D1[n]. By calculation, we obtain

that G has 18 × 2n _{– 11 edges. Also by calculation, we obtain that the edge set E(D}

1[n]) can be

divided into three partitions as follows:

E13 = {uv  E(G) | dG(u) = 1, dG(v) = 3} |E13| = 1.

E22 = {uv  E(G) | dG(u) = dG(v) = 2} |E22| = 6 × 2n– 2.

E23 = {uv  E(G) | dG(u) = 2, dG(v) = 3} |E23| = 12 × 2n – 10.

Then the edge degree partition of D1[n] is given in Table 1.

Table 1. Edge degree partition of D1[n]

dG(u) dG(v)\ uv  E(G) (1,3) (2, 2) (2, 3)

dG(e) 2 2 3

Number of edges 1 6 × 2n _{– 2 12 × 2}n _{– 10}

In the following theorem, we compute the multiplicative product connectivity Banhatti index of D1[n].

Theorem 1. The multiplicative product connectivity Banhatti index of a dendrimer nanostar D1[n] is

 





1 1 1 1

. 3

2 6 6

n

PBII D n

 

   

_  _ _  _

   

Proof: Using Table 1 and the definition of the multiplicative product connectivity Banhatti index of D1[n], we deduce

 





_{   }

ue G G

PBII D n

d u d e





   

   

 

1 1

uv E G dG u dG e dG v dG e

 

 

 

 

 



1 6 2 2 12 2 10

1 1 1 1 1 1

1 2 3 2 2 2 2 2 2 3 3 3

n

n _{ }

 

     

_  _ _  _ _  _

 

         

1 12 2 10

1 1 1 1

. 3

2 6 6

n

 

   

_  _ _  _

   

Theorem 2. The multiplicative sum connectivity Banhatti index of a dendrimer nanostar D1[n]

is

 





1 1 1 1

.

3 5 5 6

n

SBII D n

 

   

_  _ _  _

   

Proof: Using Table 1 and the definition of the multiplicative sum connectivity Banhatti index of D1[n], we deduce

 





_{ }

_{ }

1

ue G G

SBII D n

d u d e

 



 

 

 

 

uv E G d_G u d_G e d_G v d_G e

           



1 6 2 2 12 2 10

1 1 1 1 1 1

1 2 3 2 2 2 2 2 2 3 3 3

n

n _{ }

 

     

_  _ _  _ _  _

 

         

1 12 2 10

1 1 1 1

.

3 5 5 6

n

 

   

_  _ _  _

   

In the following theorem, we determine the multiplicative atom bond connectivity Banhatti index of D1[n].

Theorem 3. The multiplicative atom bond connectivity Banhatti index of a dendrimer nanostar D1[n] is

 





 

1 1 2 2 . 3 2 n n

ABCBII D n

 

   

 _  _

 

Proof: Using Table 1 and the definition of the multiplicative atom bond connectivity Banhatti index of D1[n], we deduce

 





 

_{   }

 

G G

G G

ue

d u d e

ABCBII D n

d u d e

  



 

 

   

 

   

 

G G G G

G G G G

uv E G

d u d e d v d e

d u d e d v d e

  _{   }         



1 6 2 2

1 2 2 3 2 2 2 2 2 2 2 2

1 2 3 2 2 2 2 2

12 2 10

2 3 2 3 3 2

2 3 3 3

n

 

     

_  _

 

 

 

6 2 1 _{1 2}

2 .

3 2

n

n  

  _{ }

 _  _

 

In the following theorem, we determine the multiplicative geometric-arithmetic Banhatti index of D1[n].

Theorem 4. The multiplicative geometric-arithmetic Banhatti index of a dendrimer nanostar D1[n] is

 





1

2 2 2 6 2 6

2 1 .

3 5 5

n n

GABII D n

   

   

_  _  _  _

   

Proof: Using Table 1 and the definition of the multiplicative geometric-arithmetic Banhatti index of D1[n], we deduce

 





_{ }

   

_{ }

2 G G

G G

ue

d u d e

GABII D n

d u d e







   

 

 

 

   

 

 

2 _{G G} 2 _{G G}

G G G G

uv E G

d u d e d v d e

d u d e d v d e



 

 

 

 

 

 



1 6 2 2

2 1 2 2 3 2 2 2 2 2 2 2

1 2 3 2 2 2 2 2

n

 

 _{ }   _{ } 

_  _ _  _

   

   

12 2 10

2 2 3 2 3 3

2 3 3 3

n

 

   

_  _

 

 

1 12 2 10

6 2 2

2 2 2 6 2 6

2 1 .

3 5 5

n n

   

   

_  _  _  _

   

3. DENDRIMER NANOSTARS D3[n]

In this section, we consider a family of dendrimer nanostars with n growth stages, denoted by D3[n], where n 0. The molecular graph of D3[n] with 3 growth stages is presented

Figure 2. The molecular graph of D3[3]

Let G be the graph of a dendrimer nanostar D3[n]. By calculation, we obtain that G

has 24×2n+1_{– 24 edges. Also by calculation, we obtain that the edge set E(D}

3[n]) can be divided

into four partitions:

E13 = {uv E(G) | dG(u) =1, dG(v) = 3}, |E13| = 3×2n.

E22 = {uv E(G) | dG(u) = dG(v) = 2}, |E22| = 12×2n – 6.

E23 = {uv E(G) | dG(u) = 2, dG(v) = 3}, |E23| = 24×2n – 12.

E33 = {uv E(G) | dG(u) = dG(v) = 3}, |E33| = 9×2n – 6.

Then the edge degree partition of D3[n] is given in Table 2.

Table 2. Edge degree partition of D3[n]

dG(u), dG(v)\uv E(G) (1, 3) (2, 2) (2, 3) (3, 3)

dG(e) 2 2 3 4

Number of edges 3×2n _12×2n _{– 6 24×2}n _{– 12 9×2}n _{– 6}

In the following theorems, we compute the multiplicative connectivity Banhatti indices of D3[n].

Theorem 5. The multiplicative product connectivity Banhatti index of a dendrimer nanostar D3[n] is

 





1 1 1 1 1

. 3

2 6 6 3

n n n

PBII D n

    

     

_  _ _  _ _{ }

Proof: Using Table 2 and the definition of the multiplicative product connectivity Banhatti index of D3[n], we derive

 





_{   }

ue G G

PBII D n

d u d e





   

   

 

1 1

uv E G d_G u d_G e d_G v d_G e

         



3 2 12 2 6

1 1 1 1

1 2 3 2 2 2 2 2

n n        _  _ _  _        

24 2 12 9 2 6

1 1 1 1

2 3 3 3 3 4 4 3

n n

   

   

_  _ _  _

       

3 2 24 2 12 9 2 6

1 1 1 1 1

. 3

2 6 6 3

n n n

    

     

_  _ _  _ _{ }

     

Theorem 6. The multiplicative sum connectivity Banhatti index of a dendrimer nanostar D3[n]

is

 





1 1 1 1 2

.

3 5 5 6 7

n n n

SBII D n

    

     

_  _ _  _ _{ }

     

Proof: Using Table 2 and the definition of the multiplicative sum connectivity Banhatti index of D3[n], we detive

 





_{ }

_{ }

1

ue G G

SBII D n

d u d e

 



 

 

 

 

uv E G dG u dG e dG v dG e

           



3 2 12 2 6

1 1 1 1

1 2 3 2 2 2 2 2

n n        _  _ _  _        

24 2 12 9 2 6

1 1 1 1

2 3 3 3 3 4 4 3

n n

   

   

_  _ _  _

       

3 2 24 2 12 9 2 6

1 1 1 1 2

.

3 5 5 6 7

n n n

    

     

_  _ _  _ _{ }

Theorem 7. The multiplicative atom bond connectivity Banhatti index of a dendrimer nanostar D3[n] is

 





 

3

1 2 5

2 . 3 3 2 n n n ABCBII D n

   

     

 _  _ _{ }

  _{ }

Proof: Using Table 2 and the definition of the multiplicative atom bond connectivity Banhatti index of D3[n], derive

 





 

_{   }

 

G G

G G

ue

d u d e

ABCBII D n

d u d e

  



 

 

   

 

   

 

G G G G

G G G G

uv E G

d u d e d v d e

d u d e d v d e

  _{   }         



3 2 12 2 6

1 2 2 3 2 2 2 2 2 2 2 2

1 2 3 2 2 2 2 2

n n                _  _ _  _        

24 2 12 9 2 6

2 3 2 3 3 2 3 4 2 4 3 2

2 3 3 3 3 4 4 3

n n                 _  _ _  _        

 

15 2 6 _{1 2 5}

2 .

3 3

2

n n

n    

  _{ }  

 _  _ _{ }

  _{ }

Theorem 8. The multiplicative geometric-arithmetic Banhatti index of a dendrimer nanostar D3[n] is

 





3

2 2 2 6 2 6 8 3

2 1 .

3 5 5 7

n n n

n GABII D n

    

 

     

_  _  _  _ _{ }

     

Proof: Using Table 2 and the definition of the multiplicative geometric-arithmetic Banhatti index of D3[n], we derive

 





_{ }

   

_{ }

2 _{G G}

G G

ue

d u d e

GABII D n

d u d e

 



   

 

 

 

   

 

2 _{G G} 2 _{G G}

G G G G

uv E G

d u d e d v d e

d u d e d v d e

            



3 2 12 2 6

2 1 2 2 3 2 2 2 2 2 2 2

1 2 3 2 2 2 2 2

24 2 12 9 2 6

2 2 3 2 3 3 2 3 4 2 4 3

2 3 3 3 3 4 4 3

n n

   

       

_  _ _  _

   

   

3 2 24 2 12 9 2 6

12 2 6

2 2 2 6 2 6 8 3

2 1 .

3 5 5 7

n n n

n

    

 

     

_  _  _  _ _{ }

     

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