**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

###

###

_{ }

###

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 ****D****1[****n****]**

In this section, we consider a family of dendrimer nanostars with *n* growth stages,
denoted by *D*1[*n*], where *n*0. The molecular graph of *D*1[*n*] with 4 growth stages is depicted

in Figure 1.

Let *G* be the molecular graph of dendrimer nanostar *D*1[*n*]. By calculation, we obtain

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

1[*n*]) can be

divided into three partitions as follows:

*E*13 = {*uv ** E*(*G*) | *dG(u*) = 1, *dG(v*) = 3} |*E*13| = 1.

*E*22 = {*uv ** E*(*G*) | *dG(u*) = *dG(v*) = 2} |*E*22| = 6 × 2*n*– 2.

*E*23 = {*uv ** E*(*G*) | *dG(u*) = 2, *dG(v*) = 3}* * |*E*23| = 12 × 2*n* – 10.

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

**Table 1. Edge degree partition of ****D****1[****n****] **

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

*dG*(*e*) 2 2 3

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

In the following theorem, we compute the multiplicative product connectivity
Banhatti index of *D*1[*n*].

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

###

###

1###

1 12 2 101 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 *D*1[*n*], we deduce

###

###

1###

_{ }

1
*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 *D*1[*n*]

is

###

###

1###

1 12 2 101 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 *D*1[*n*], we deduce

###

###

1###

_{ }

_{ }

1

*ue* *G* *G*

*SBII D n*

*d* *u* *d* *e*

###

###

###

###

###

1 1*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 *D*1[*n*].

**Theorem 3**. The multiplicative atom bond connectivity Banhatti index of a dendrimer
nanostar *D*1[*n*] is

###

###

###

##

6 2 1 12 2 101
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 *D*1[*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

12 2 10

2 3 2 3 3 2

2 3 3 3

*n*

_{} _{}

##

12 2 106 2 1 _{1} _{2}

2 .

3 2

*n*

*n*

_{} _{}

_{} _{}

In the following theorem, we determine the multiplicative geometric-arithmetic
Banhatti index of *D*1[*n*].

**Theorem 4**. The multiplicative geometric-arithmetic Banhatti index of a dendrimer nanostar
*D*1[*n*] is

###

###

###

1 6 2 2 12 2 101

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 *D*1[*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 ****D****3[****n****] **

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

**Figure 2. The molecular graph of ****D****3[3] **

Let* G *be the graph of a dendrimer nanostar *D*3[*n*]. By calculation, we obtain that *G *

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

3[*n*]) can be divided

into four partitions:

*E*13 = {*uv** E*(*G*) | *dG(u*) =1, *dG(v*) = 3}, |*E*13| = 3×2*n*.

*E*22 = {*uv** E*(*G*) | *dG(u*) = *dG(v*) = 2}, |*E*22| = 12×2*n* – 6.

*E*23 = {*uv** E*(*G*) | *dG(u*) = 2, *dG(v*) = 3}, |*E*23| = 24×2*n* – 12.

*E*33 = {*uv** E*(*G*) | *dG(u*) = *dG(v*) = 3}, |*E*33| = 9×2*n* – 6.

Then the edge degree partition of *D*3[*n*] is given in Table 2.

**Table 2. Edge degree partition of ****D****3[****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 3×2*n* _{12×2}*n _{ – }*

_{6 }

_{24×2}

*n*

_{ – }_{12 }

_{9×2}

*n*

_{ – }_{6 }

In the following theorems, we compute the multiplicative connectivity Banhatti
indices of *D*3[*n*].

**Theorem 5.** The multiplicative product connectivity Banhatti index of a dendrimer nanostar
*D*3[*n*] is

###

###

1###

3 2 24 2 12 9 2 61 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 *D*3[*n*], we derive

###

###

3###

_{ }

1
*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 *D*3[*n*]

is

###

###

3###

3 2 24 2 12 9 2 61 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 *D*3[*n*], we detive

###

###

3###

_{ }

_{ }

1

*ue* *G* *G*

*SBII D n*

*d* *u* *d* *e*

###

###

###

###

###

1 1*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 *D*3[*n*] is

###

###

###

##

15 2 6 24 2 12 9 2 63

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 *D*3[*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 2 615 2 6 _{1} _{2} _{5}

2 .

3 3

2

*n*
*n*

*n*

_{} _{}

_{} _{} _{} _{}

_{} _{}

**Theorem 8**. The multiplicative geometric-arithmetic Banhatti index of a dendrimer nanostar
*D*3[*n*] is

###

###

###

3 2 12 2 6 24 2 12 9 2 63

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 *D*3[*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

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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