# Multiplicative Connectivity Banhatti Indices of Dendrimer Nanostars

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### Dendrimer Nanostars

V. R. Kulli

Department of Mathematics,

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

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.

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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

2 .

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

2

G

G

.

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.

(3)

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)\ uvE(G) (1,3) (2, 2) (2, 3)

dG(e) 2 2 3

Number of edges 1 6 × 2n – 2 12 × 2n – 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 12 2 10

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

1

1

ue G G

PBII D n

d u d e

###    

 

1 1

uv E GdG 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

 

   

   

(4)

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

is

1

### 

1 12 2 10

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

1

ue G G

SBII D n

d u d e

 

###  

  1 1

uv E GdG 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 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

##  

6 2 1 12 2 10

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

1

2

G G

G G

ue

d u d e

ABCBII D n

d u d e

  

###  

  2 2

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

(5)

12 2 10

2 3 2 3 3 2

2 3 3 3

n

 

     

 

 

##  

12 2 10

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 6 2 2 12 2 10

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

1

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

(6)

Figure 2. The molecular graph of D3

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 12×2n6 24×2n12 9×2n6

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

### 

3 2 24 2 12 9 2 6

1 1 1 1 1

. 3

2 6 6 3

n n n

PBII D n

    

     

(7)

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

3

1

ue G G

PBII D n

d u d e

###    

 

1 1

uv E GdG 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 1

. 3

2 6 6 3

n n n

    

     

     

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

is

3

### 

3 2 24 2 12 9 2 6

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

3

1

ue G G

SBII D n

d u d e

 

###  

  1 1

uv E GdG 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

    

     

(8)

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

##  

15 2 6 24 2 12 9 2 6

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

3

2

G G

G G

ue

d u d e

ABCBII D n

d u d e

  

###  

  2 2

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                         

##  

24 2 12 9 2 6

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 12 2 6 24 2 12 9 2 6

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

3

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

(9)

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

    

 

     

 

     

REFERENCES

1. V.R.Kulli, College Graph Theory, Vishwa International Publications, Gulbarga, India (2012).

2. Gutman and O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin (1986).

3. V.R.Kulli, Multiplicative connectivity indices of certain nanotubes, Annals of Pure and Applied Mathematics, 12(2), 169-176 (2016).

4. V.R.Kulli, Multiplicative connectivity Banhatti indices of benzenoid system and polycyclic aromatic hydrocarbons, submitted.

5. V.R. Kulli, First multiplicative K Banhatti index and coindex of graphs, Annals of Pure and Applied Mathematics, 11(2), 79-82 (2016).

6. V.R. Kulli, Second multiplicative K Banhatti index and coindex of graphs, Journal of Computer and Mathematical Sciences, 7(5), 254-258 (2016).

7. V.R.Kulli, On K-hyper-Banhatti indices and coindices of graphs, International Research Journal of Pure Algebra, 6(5), 300-304 (2016).

8. V.R. Kulli, On multiplicative K-Banhatti and multiplicative K hyper-Banhatti indices of V-Phenylenic nanotubes and nanotorus, Annals of Pure and Applied Mathematics, 11(2), 145-150 (2016).

9. V.R. Kulli, Multiplicative K hyper-Banhatti indices and coindices of graphs, International Journal of Mathematical Archive, 7(6), 60-65 (2016).

10. V.R.Kulli, New K Banhatti topological indices, International Journal of Fuzzy Mathematical Archive, 12(1) 29-37 (2017).

11. V.R.Kulli, A new Banhatti geometric-arithmetic index, International Journal of Mathematical Archive, 8(4), 112-115 (2017).

12. V.R.Kulli, B. Chaluvaraju and H.S. Boregowda, Connectivity Banhatti indices for certain families of benzenoid systems, Journal of Ultra Chemistry, 13(4), 81-87 (2017).

13. V.R.Kulli, B.Chaluvaraju and H.S. Baregowda, K-Banhatti and K hyper-Banhatti indices of windmill graphs, South East Asian J. of Math. and Math. Sci, 13(1), 11-18 (2017). 14. V.R.Kulli, Two new multiplicative atom bond connectivity indices, Annals of Pure and

Applied Mathematics, 13(1), 1-7 (2017).

15. V.R.Kulli, Multiplicative connectivity indices of nanostructures, Journal of Ultra Scientist of Physical Sciences, A 29(1), 1-10 (2017).

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17. V.R.Kulli, Some new multiplicative geometric-arithmetic indices, Journal of Ultra Scientist of Physical Sciencs, A, 29(2), 52-57 (2017).

18. V.R.Kulli, New multiplicative inverse sum indeg index of certain benzoid systems, Journal of Global Research in Mathematical Archieves, 4(10), 15-19 (2017).

19. V.R.Kulli, New multiplicative arithmetic-geometric indices, Journal of Ultra Scientist of Physical Sciences, A, 29(6), 205-211 (2017).

20. V.R. Kulli, Multiplicative Banhatti and multiplicative hyper-Banhatti indices of certain networks, Journal of Computer and Mathematical Sciences, 8(12), 750-757 (2017). 21. V.R.Kulli, Edge version of multiplicative connectivity indices of some nanotubes and

nanotorus, International Journal of Current Research in Science and Technology, 3(11), 7-15 (2017).

22. V.R.Kulli, B. Stone, S. Wang and B.Wei, Generalized multiplicative indices of polycyclic aromatic hydrocarbons and benzenoid systems, Z. Naturforsch, 72(6) a, 573-576 (2017). 23. N.M. Husin, R. Hasni and N.E. Arif, Atom bond connectivity and geometric-arithmetic

Figure   References

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