Multiplicative Connectivity Banhatti Indices of Benzenoid Systems and Polycyclic Aromatic Hydrocarbons

(1)

ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

Multiplicative Connectivity Banhatti Indices of Benzenoid Systems and Polycyclic Aromatic Hydrocarbons

V. R. Kulli

Department of Mathematics,

Gulbarga University, Gulbarga 585106, INDIA.

email: vrkulli @gmail.com.

(Received on: February 26, 2018)

ABSTRACT

The Zagreb and K Banhatti indices are closely related. Recently, the author introduced the multiplicative connectivity indices of a graph. In this paper, we propose some multiplicative connectivity Banhatti indices of a graph. Also we determine the multiplicative product connectivity Banhatti index, multiplicative sum connectivity Banhatti index, multiplicative atom bond connectivity Banhatti index and multiplicative geometric-arithmetic Banhatti index of certain important chemical structures like benzenoid systems and polycyclic aromatic hydrocarbons.

Mathematics Subject Classification: 05C05, 05C07, 05C90.

Keywords: multiplicative product connectivity Banhatti index, multiplicative sum connectivity Banhatti index, multiplicative ABC Banhatti index, multiplicative geometric-arithmetic Banhatti index, benzenoid system, polycyclic aromatic hydrocarb.

1. INTRODUCTION

In this paper, we consider only finite, connected, simple graph G with vertex set V(G) and edge set E(G). The degree d

G

(v) of a vertex v is the number of vertices adjacent to v. The edge e connecting the vertices u and v denoted by uv. If e=uv is an edge of G, then the vertex u and edge e are incident as are v and e. Let d

G

(e) denote the degree of an edge e in G, which is defined by d

G

(e) = d

G

(u) + d

G

(v) – 2 with e=uv. We refer to

¹

for other undefined term and notation.

A molecular graph is a graph in which the vertices correspond to the atoms and the

edges to the bonds of a molecule. A single number that can be computed from the molecular

graph and used to characterize some property of the underlying molecule is said to be a

(2)

topological index. Numerous such topological indices have been considered in theoretical chemistry, and have found some applications, especially in QSPR/QSAR research, see

²

.

In

³

, Kulli introduced the first and second K Banhatti indices, intending to take into account the contributions of pairs of incident elements. The first and second K Banhatti indices of a graph G are defined as

     

1 G G

ue

B G     d u  d e   ^and

²

^{ }

^G

^{   }

^G

ue

B G   d u d e where ue means that the vertex u and edge e are incident in G.

Motivated by the definition of the multiplicative atom bond connectivity index

⁴

and its wide applications, we introduce the multiplicative product connectivity Banhatti index, multiplicative sum connectivity Banhatti index, multiplicative atom bond connectivity Banhatti index and multiplicative geometric-arithmetic index of a molecular graph as follows:

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

     

1 .

ue G G

PBII G

d u d e

  ⁽¹⁾

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

     

1 .

ue G G

SBII G

d u d e

   ⁽²⁾

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

     

    ² ^.

G G

ue

d u d e

ABCBII G

d u d e

 

  ⁽³⁾

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

   ⁽⁴⁾

Recently, many K Banhatti indices were studied, for example, in

5,6,7,8,9,10,11,12,13

. Also some multiplicative connectivity indices were studied, for example, in

14,15,16,17,18,19,20,21

.

In this paper, we determine the multiplicative connectivity Banhatti indices for jagged- rectangle benzenoid systems B

m,n

and polycyclic aromatic hydrocarbons PAH

n

. For more information about jagged-rectangle benzenoid systems and polycyclic aromatic hydrocarbons see

²²

.

2. RESULTS FOR BENZENOID SYSTEMS

In this section, we focus on the chemical graph structure of a jagged-rectangle

benzenoid system, symbolized by B

m,n

, for m, n  N. Three chemical graphs of a jagged-

rectangle benzenoid system are depicted in Figure 1.

(3)

Figure 1. Chemical graphs of a jagged-rectangle benzenoid system

Let G be the graph of a jagged-rectangle benzenoid system B

m,n

. By calculation, we obtain that G has 4mn+4m+2n – 2 vertices and 6mn+5m+n – 4 edges, see [22]. In G, the edge set E(B

m,n

) can be divided into three partitions based on the degree of end vertices of each edge as follows:

E

22

= {uv E(G) | d

G

(u) = d

G

(v) = 2}, |E

22

| = 2n + 4.

E

23

= {uv E(G) | d

G

(u) = 2, d

G

(v) = 3}, |E

23

| = 4m + 4n – 4.

E

33

= {uv E(G) | d

G

(u) = d

G

(v) = 3}, |E

33

| = 6mn + m – 5n – 4.

Then the edge degree partition of B

m,n

is given in Table 1.

Table 1. Edge degree partition of Bm,n

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

dG(e) 2 3 4

Number of edges 2n+4 4m+4n – 4 6mn+m – 5n – 4

In the following theorem, we compute the multiplicative product connectivity Banhatti index of B

m, n

.

Theorem 1. Let B

m, n

be the family of a jagged-rectangle benzenoid system. Then

 

^,

¹ ¹ ₃

⁴ ⁴ ⁴

¹

⁶ ⁵ ⁴

^.

6 3

m n mn m n

PBII B

m n

    

   

          

Proof: Let G be the graph in the family of a jagged-rectangle benzenoid system B

m,n

. Using equation (1) and Table 1, we derive

 

^{m n}^,

_{   } ¹

ue G G

PBII B

d u d e

 

       

 

1 1

uv E G

d

G

u d

G

e d

G

v d

G

e

 

 

 

 

 



(4)

4 4 4 6 5 4 2 4

1 1 1 1 1 1

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

m n mn m n

n     

   

 

                       

4 4 4 6 5 4

1 1 1

3 .

6 3

m n mn m  n

   

          

In the following theorem, we determine the multiplicative sum connectivity Banhatti index of B

m, n

.

Theorem 2. Let B

m,n

be the family of a jagged-rectangle benzenoid system. Then

 

^,

¹ ¹

⁴ ⁴ ⁴

²

⁶ ⁵ ⁴

^.

5 6 7

m n mn m n

SBII B

m n

    

   

          

Proof: Let G be the graph in the family of a jagged-rectangle benzenoid system B

m,n

. Using equation (2) and Table 1, we derive

 

^{m n}^,

_{ } ¹ _{ }

ue G G

SBII B

d u d e

  

       

 

1 1

uv E G

d

G

u d

G

e d

G

u d

G

e

 

 

 

   

 



4 4 4 6 5 4

2 4

1 1 1 1 1 1

2 2 2 2 2 3 3 3 3 4 3 4 .

m n mn m n

n     

   

 

                       

4 4 4 6 5 4

1 1 2

5 6 7 .

m n mn m  n

   

          

In the following theorem, we determine the multiplicative atom bond connectivity Banhatti index of B

m, n

.

Theorem 3. Let B

m,n

, be the family of a jagged-rectangle benzenoid system. Then

 

^{m n}^,

  ²

⁴ⁿ ⁴

¹ ₂ ² ₃

⁴^m ⁴ⁿ ⁴

⁵ ₃

⁶^{mn m} ⁵ⁿ ⁴

^.

ABCBII B

  



 

 

 

           

Proof: Let G be the graph in the family of a jagged-rectangle benzenoid system B

m,n

. By using equation (3) and Table 1, we deduce

 

^{m n}^, ^G

^{ } _{   }

^G

^{ } ²

G G

ue

d u d e

ABCBII B

d u d e

 

 

(5)

   

       

   

 

2 2

G G G G

uv E G

d u d e d v d e



     

   

 

 



2 4 4 4 4

2 2 2 2 2 2 2 3 2 3 3 2

2 2 2 2 2 3 3 3

n m n

           

               

6 5 4

3 4 2 3 4 2

3 4 3 4 .

mn m  n

     

       

  ²

²ⁿ^⁴

^ ¹ ₂ ² ₃ ^

⁴^m^{ }⁴ⁿ ⁴

^ ⁵ ₃ ^

⁶^{mn m}^{  }⁵ⁿ ⁴

^.

           

In the following theorem, we determine the multiplicative geometric-arithmetic Banhatti index of B

m, n

.

Theorem 4. Let B

m,n

, be the family of a jagged-rectangle benzenoid system. Then

 

^{m n}^,

²

⁴ⁿ ⁴

^{2 6} ₅ ¹

⁴^m ⁴ⁿ ⁴

^{8 3} ₇

⁶^mn ⁵^{m n} ⁴

^.

GAII B

    



   

           

Proof: Let G be the graph in the family of a jagged-rectangle benzenoid system B

m,n

. By using equation (4) and Table 1, we deduce

  _{ } ^{   } _{ } _{ } ^{   } _{ } _{ } ^{   } _{ }

,  

2

_G _G

2

_G _G

2

_G _G

m n

G G G G G G

ue uv E G

d u d v d u d e d v d e

GABII B

d u d v

_

d u d e d v d e

 

 

  

      

 

2 4 4 4 4

2 2 2 2 2 2 2 2 3 2 3 3

2 2 2 2 2 3 3 3

n m n

       

               

6 5 4

2 3 4 2 3 4

3 4 3 4 .

mn m  n

   

       

4 4 4 6 5 4

2 4

2 6 8 3

2 1 .

5 7

m n mn m n

n

    



   

           

3. RESULTS FOR POLYCYCLIC AROMATIC HYDROCARBONS

In this section, we consider a family of polycyclic aromatic hydrocarbons, denoted by

PAH

n

. The first three members of the family of polycyclic aromatic hydrocarbons are

presented in Figure 2.

(6)

Figure 2. First three elements of PAHn

Let G be the molecular graph in the family of polycylic aromatic hydrocarbons PAH

n

. By calculation, we obtain that G has 9n

²

+3n edges, see

²²

. In G, the edge set E(PAH

n

) can be divided into two partitions based on the degree of end vertices of each edge as follows:

E

13

= {uv E(G) | d

G

(u) = 1, d

G

(v) = 3}, |E

13

| = 6n.

E

33

= {uv E(G) | d

G

(u) = d

G

(v) = 3}, |E

33

| = 9n

²

– 3n.

Then the edge degree partition of PAH

n

is given in Table 2.

Table 2. Edge degree partition of PAHn

d

G

(u), d

G

(v)\uv E(G) (1, 3) (3, 3)

d

G

(e) 2 4

Number of edges 6n 9n

²

– 3n

In the following theorem, we compute the multiplicative product connectivity Banhatti index of PAH

n

.

Theorem 5. Let PAH

n

be the family of polycyclic aromatic hydrocarbons. Then

 

6 9 2 3

1 1 1

2 6 3 .

n n n

PBII PAH

n

   



          

Proof: Let G be the graph in the family of polycyclic aromatic hydrocarbons PAH

n

. Using equation (1) and Table 2, we deduce

 

   

1

n

ue G G

PBII PAH

d u d e

 

       

 

1 1

uv E G

d

G

u d

G

e d

G

v d

G

e

 

 

 

   

 



6 9 2 3

1 1 1 1

1 2 3 2 3 4 3 4

n n n

   

               

(7)

6 9 2 3

1 1 1

2 6 3 .

n n  n

   

          

In the next theorem, we compute the multiplicative sum connectivity Banhatti index of PAH

n

. Theorem 6. Let PAH

n

be the family of polycyclic aromatic hydrocarbons. Then

 

6 9 2 3

1 1 2

3 5 7 .

n n n

SBII PAH

n

   



          

Proof: Let G be the graph in the family of polycyclic aromatic hydrocarbons PAH

n

. Using equation (2) and Table 2, we deduce

 

   

1

n

ue G G

SBII PAH

d u d e

  

       

 

1 1

uv E G

d

G

u d

G

e d

G

v d

G

e

 

 

 

   

 



6 9 2 3

1 1 1 1

1 2 3 2 3 4 3 4 .

n n n

   

               

6 9 2 3

1 1 2

3 5 7 .

n n n

   

          

In the following theorem, we compute the multiplicative atom bond connectivity Banhatti index of PAH

n

.

Theorem 7. Let PAH

n

be the family of polycyclic aromatic hydrocarbons. Then



ⁿ

   ²

⁶ⁿ

⁵ ₃

⁹ⁿ² ³ⁿ

^.

ABCBII PAH

 



     

Proof: Let G be the graph in the family of polycyclic aromatic hydrocarbons PAH

n

. Using equation (3) and Table 2, we deduce



n



^G

^{ } _{   }

^G

^{ } ²

G G

ue

d u d e

ABCBII PAH

d u d e

 

 

   

       

   

 

2 2

G G G G

uv E G

d u d e d v d e



     

   

 

 



(8)

6 9 2 3

1 2 2 3 2 2 3 4 2 3 4 2

1 2 3 2 3 4 3 4

n n  n

           

               

  ²

⁶ⁿ

^ ⁵ ₃ ^

⁹ⁿ²^³ⁿ

^.

     

In the following theorem, we compute the multiplicative geometric-arithmetic Banhatti index of PAH

n

.

Theorem 8. Let PAH

n

be the family of polycyclic aromatic hydrocarbons. Then

 

6 9 2 3

2 2 2 6 8 3

3 5 7 .

n n n

GAB PAH

n

   



          

Proof: Let G be the graph in the family of polycyclic aromatic hydrocarbons PAH

n

. By using equation (4) and Table 2, we deduce

  _{ } ^{   } _{ } _{ } ^{   } _{ } _{ } ^{   } _{ }

 

2

_G _G

2

_G _G

2

_G _G

n

G G G G G G

ue uv E G

d u d v d u d e d v d e

GABII PAH

d u d v

_

d u d e d v d e

 

 

  

      

 

5 9 2 3

2 1 2 2 3 2 2 3 4 2 3 4

1 2 3 2 3 4 3 4 .

n n 

       

               

6 9 2 3

2 2 2 6 8 3

3 5 7 .

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, K-Banhatti indices of graphs, Journal of Computer and Mathematical Sciences, 7(4), 213-218 (2016).

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

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).

(9)

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).

16. V.R.Kulli, A new multiplicative arithmetic-geometric index, International Journal of Fuzzy Mathematical Archive, 12(2), 49-53 (2017).

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), 573-576 a (2017).

Multiplicative Connectivity Banhatti Indices of Benzenoid Systems and Polycyclic Aromatic Hydrocarbons