# Some New Multiplicative Connectivity Kulli-Basava Indices

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## Kulli-Basava Indices

V.R.Kulli

Department of Mathematics.

Gulbarga University, Gulbarga 585106, India

Abstract: A topological index for a graph is used to determine some property of the graph of molecular by a single number. In this paper, we introduce the multiplicative sum connectivity Kulli-Basava index, multiplicative product connectivity Basava index, multiplicative ABC Basava index and multiplicative GA Kulli-Basava index of a graph. We compute these multiplicative connectivity Kulli-Kulli-Basava indices of regular, wheel and helm graphs.

Keywords: Multiplicative sum connectivity Basava index, multiplicative product connectivity Kulli-Basava index, multiplicative ABC Kulli-Kulli-Basava index, multiplicative GA Kulli-Kulli-Basava index, graph.

Mathematics Subject Classification: 05C05, 05C07, 05C12, 05C35.

I. Introduction

Let G be a finite, simple, connected graph with 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. The edge connecting the vertices u and v will be denoted by

uv. The degree of an edge e=uv in a graph G is defined by dG(e)= dG(u)+ dG(v) – 2. Let Se(v) denote the sum of the degrees of all edges incident to a vertex v. For undefined term and notation, we refer . Topological indices or graph indices have their applications in various disciplines of Science and Technology.

In , Kulli introduced the first and second multiplicative Kulli-Basava indices of a graph, defined as

 

1 ,

 

ee

uv E G

KB II G S u S v

 

2 e e .

uv E G

KB II G S u S v

 

### 

Recently, the connectivity Kulli-Basava indices , square Kulli-Basava index , multiplicative F -Kulli-Basava index , first and second hyper -Kulli-Basava indices , F-Kulli-Basava index  multiplicative hyper Kulli-Basava indices  were introduced and studied.

We introduce the multiplicative sum connectivity Kulli-Basava index and multiplicative product connectivity Kulli-Basava index of a graph, defined as

 

1

,

uv E G e e

SKBII G

S u S v

 

1 .

### 

uv E G e e

PKBII G

S u S v

Also we propose the multiplicative atom bond connectivity Kulli-Basava index, multiplicative geometric-arithmetic Kulli-Basava index and multiplicative reciprocal Kulli-Basava index of a graph, defined as

 

2 ,

  

e e

uv E G e e

S u S v

ABCKBII G

S u S v

 

2

,

e e

uv E G e e

S u S v GAKBII G

S u S v

 

.

### 

e e

uv E G

RKBII G S u S v

Recently, some connectivity indices were studied [ 9, 10, 11, 12, 14, 15, 16,17].

Finally, we introduce the general first and second Kulli-Basava indices of a graph, defined as

 

1 ,

 

a a

e e

uv E G

(2)

 

2 ,

a a

e e

uv E G

KB II G S u S v

 

### 

where a is a real number.

Recently some multiplicative topological indices were studied [18, 19, 20, 21].In this paper, some multiplicative connectivity Kulli-Basava indices of regular, wheel, helm graphs are computed.

II. Results for regular graphs

A graph G is an r-regular graph if the degree of every vertex of G is r.

Theorem 1. Let G be an r-regular graph with n vertices and m edges. Then

 

### 

1 4 1 .

am a

KB II Gr r

(1)

Proof: If G is an r-regular graph with n vertices and m edges, then Se(u) = 2r(r – 1) for any vertex u in G. Thus,

 

   

1 2 1 2 1

 

### 

a    am a

e e

uv E G

KB II G S u S u r r r r

 

4 1

### 

  am

r r

Corollary 1.1. If G is an r-regular graph with n vertices and m edges, then

###  

 

1

.

4 1

 

 

 

m

SKBII G

r r

Proof: Put a = –½ in equation (1), we get the desired result.

Corollary 1.2. If Kn is a complete graph with n vertices, then

### 

 1 2

1

.

2 1 2

 

 

 

n n

n

SKBII K

n n

Proof: Put r = n – 1,

1

### 

2

 n n

m and a = 1 2

 in equation (1), we get the desired result.

Corollary 1.3. If Cn is a cycle with n vertices, then

###  

1 . 2 2

   

n

n

SKBII C

Proof: Put r = 2, m = n and a = 1 2

 in equation (1), we get the desired result.

Theorem 2. Let G be an r-regular graph with n vertices and m edges. Then

  2 2

2 4 1 .

am a

KB II G  r r  (2)

Proof: Let G be an r-regular graph with n vertices and m edges. Then Se(u) = 2r(r – 1) for any vertex u in G. Therefore

 

   

### 

2 2 1 2 1

am a

a

e e

uv E G

KB II G S u S v r r r r

 

### 

   2 2

4 1

am

r r

 

  

Corollary 2.1. If G is an r-regular graph with n vertices and m edges, then

###  

1  .

2 1

m

PKBII G

r r

 

 

Proof: Put a = –½ in equation (2), we get the desired result.

Corollary 2.2. If Kn is a complete graph with n vertices and m edges, then

### 

 1 2

1

.

2 1 2

n n

n

PKBII K

n n

 

 

Proof: Put r = n – 1,

1

### 

2

 n n

m and a = 1 2

 in equation (2), we get the desired result.

(3)

###  

1 . 4

n

n

PKBII C    

 

Proof: Put r = 2, m = n and a = 1 2

 in equation (2), we get the desired result.

Corollary 2.4. If G is an r-regular graph with n vertices and m edges, then

2  1

### 

m.

RKBII Gr r

Proof: Put a = ½ in equation (2), we get the desired result.

Corollary 2.5. If Kn is a complete graph with n vertices and m edges, then

### 

 1 2

1

.

2 1 2

n n

n

RKBII K

n n

 

 

Proof: Put a = 1 2

 , r = n – 1

1

### 

2

 n n

m in equation (2), we get the desired result.

Corollary 2.6. If Cnis a cycle with n vertices and m edges, then

###  

4 .n n

RKBII C

Proof: Put r = 2, m = n, a = 1 2

 , in equation (2), we get the desired result.

Theorem 3. Let G be an r-regular graph with n vertices and m edges. Then

###  

 

 

4 1 2

.

2 1

m

r r ABCKBII G

r r

 

  

 

Proof: If G is an r-regular graph with n vertices and m edges, then Se(u) = 2r(r – 1) for any vertex u in G. Therefore

 

2

e e

uv E G e e

S u S v

ABCKBII G

S u S v

  

### 

       

   

2 1 2 1 2 4 1 2

2 1 2 1 2 1

m m

r r r r r r

r r r r r r

       

   

    

 

Theorem 4. Let G be an r-regular graph with n vertices and m edges. Then

GAKBII(G) = 1.

Proof: If G is an r-regular graph with n vertices and m edges, then Se(u) = 2r(r – 1) for any vertex u in G. Thus

 

2 e e

uv E G e e

S u S v GAKBII G

S u S v

### 

       

2 2 1 2 1

1

2 1 2 1

m

r r r r

r r r r

  

  

 

III. Results for wheel graphs

A wheel Wn is the join of Cn and K1. Then Wn has n+1 vertices and 2n edges. The vertices of Cn are called rim vertices and the vertex K1 is called apex.

Lemma 5. Let Wn be a wheel with n+1 vertices and 2n edges. Then

E1 = {uvE(G) | Se(u) = n(n+1), Se(v) = n+9}, |E1| = n.

E2 = {uvE(G) | Se(u) = Se(v) = n + 9}, |E2| = n.

Theorem 6. Let Wn be a wheel with n+1 vertices and 2n edges, n3. Then

2

1 2 9 2 18 .

an an

a n

KB II Wnn  n

(3)

(4)

 

### 

1 1 9 9 9

n

an an

a a

n e e

uv E W

KB II W S u S v n n n n n

 

       

2

### 

2 9 an 2 18 an

n n n

    

Corollary 6.1. If Wn is a wheel graph with n+1 vertices and 2n edges, then

###  

2

1 1

.

2 18

2 9

n n

n

SKBII W

n

n n

   



 

 

 

Proof: Put a = 1 2

 in equation (3), we get the desired result.

Theorem 7. Let Wn be a wheel with n+1 vertices and 2n edges, n3. Then

### 

3

2 1 9 .

an an

a n

KB II Wn n  n

(4)

Proof: Let Wn be a wheel with n+1 vertices and 2n edges. By using definition and Lemma 5, we derive

 

### 

2 1 9 9 9

n

an an

a a

n e e

uv E W

KB II W S u S v n n n n n

 

     

### 

3

1 an 9 an.

n n n

   

Corollary 7.1. Let Wn be a wheel with n+1 vertices and 2n edges. Then

i)

3 2

1 1

. 9 1

n n

n

PKBII W

n n n

     

   

 

ii)

 

### 

 

3 2

1 9 .

n n

n

RKBII Wn n  n

Proof: Put a = 1, 2

 1

2 in equation (4), we get the desired results.

Theorem 8. Let Wn be a wheel with n vertices and 2n edges. Then

2

2

### 

2 2

2 7 2 16

1 7 9

n n

n

n n n

ABCKBII W

n n n n

       

 

    

Proof: By using definition and Lemma 5, we obtain

 

2

e e

n

uv E G e e

S u S v

ABCKBII W

S u S v

  

### 

1 9 2 9 9 2

1 9 9 9

n n

n n n n n

n n n n n

           

   

   

   

### 

2 2 2

2

2 7 2 16

.

1 9 9

n n

n n n

n n n n

       

 

    

Theorem 9. Let Wn be a wheel with n+1 vertices and 2n edges. Then

2

2 1

9

### 

.

2 9

n

n

n n n

GAKBII W

n n

  

 

 

Proof: By using definition and Lemma 5, we obtain

 

2

n

e e

n

uv E W e e

S u S v GAKBII W

S u S v

(5)

2 1 9 2 9 9

1 9 9 9

n n

n n n n n

n n n n n

 

   

     

   

### 

2

2 1 9

.

2 9

n

n n n

n n

  

 

 

IV. Results for helm graphs

A helm graph Hn is a graph obtained from Wn by attaching an end edge to each rim vertex. We see that

Hn has 2n+1 vertices and 3n edges.

Lemma 10. Let Hn be a helm graph with 2n+1 vertices and 3n edges. Then Hn has three types of edges as E1 = {uvE(Hn) | Se(u) = n(n+2), Se(v) = n+17}, |E1| = n.

E2 = {uvE(Hn) | Se(u) = Se(v) = n + 17}, |E2| = n.

E3 = {uvE(Hn) | Se(u) = n+17, Se(v) = 3}, |E3| = n.

Theorem 11. Let Hn be a helm graph with 2n+1 vertices and 3n edges. Then

2

### 

1 3 17 2 34 20 .

an an an

a n

KB II Hnn  n  n

(5)

Proof: Let Hn be a helm graph with 2n+1 vertices and 3n edges. Then

 

2

n

a a

n e e

uv E H

KB II H S u S v

 

n n 1 n 17

an

n 17

n 17

an

n 17 3

### 

an

          

n2 3n 17

an

2n 34

an

n 20

### 

an

      

Corollary 11.1. If Hn is a helm graph with 2n+1 vertices and 3n edges, then

###  

2 2 2

2

1 1 1

3 17 2 34 20

n n n

n

SKBII H

n n n n

     

   

     

Proof: Put a = 1 2

 in equation (5), we get the desired result.

Theorem 12. Let Hn be a helm graph with 2n+1 vertices and 3n edges. Then

2

### 

2 2 17 17 3 17 .

an an an

a n

KB II Hn nn  nn

(6)

Proof: By using definition and Lemma 10, we deduce

 

2

n

a a

n e e

uv E H

KB II H S u S v

 

n n 2 n 17

an

n 17

n 17

an

n 17 3

an

       

2

### 

2 17 an 17 an 3 17 an.

n n n n n

      

Corollary 12.1. Let Hn be a helm graph with 2n+1 vertices and 3n edges. Then

i)

1

2 1

1

2.

2 17 17 17

n n

n

n

PKBII H

n n n n n n

     

     

   

ii)

1 17

2  17

317

### 

2.

n n

n n

RKBII Hn nn  nn

Proof: Put a = 1, 2

 1

2 in equation (6), we get the desired results.

Theorem 13. Let Hn be a helm graph with 2n+1 vertices and 3n edges. Then

2

2

2

### 

2 2

3 15 2 32 18

2 17 17 3 17

n n n

n

n n n n

ABCKBII H

n n n n n

          

  

      

(6)

 

2

n

e e

n

uv E H e e

S u S v

ABCKBII H

S u S v

  

### 

2 2 2

2 17 2 17 17 2 17 3 2

2 17 17 17 3 17

n n n

n n n n n n

n n n n n n

                

    

     

### 

2 2 2 2

2

3 15 2 32 18

.

2 17 17 3 17

n n n

n n n n

n n n n n

          

  

      

Theorem 14. If Hn is a helm graph with 2n+1 vertices and 3n edges, then

2

2 2

17

2 3

17

### 

.

3 17 20

n n

n

n n n n

GAKBII H

n n n

 

   

  

   

Proof: By using definition and Lemma 5, we obtain

 

2

n

e e

n

uv E H e e

S u S v GAKBII H

S u S v

### 

2 2 17 2 7 17 2 17 3

2 17 17 17 17 3

n n n

n n n n n n

n n n n n n

   

     

           

 

### 

2

2 2 17 2 3 17

.

3 17 20

n n

n n n n

n n n

 

   

  

   

References

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

 V.R. Kulli, Multiplicative Kulli-Basava and multiplicative hyper Kulli-Basava indices of some graphs, International Journal of Mathematical Archive, 10(8) (2019) 18-24.

 V.R. Kulli, New connectivity topological indices, Annals of Pure and Applied Mathematics, 20(1) (2019) 1-8.

 V.R. Kulli, Some new Kulli-Basava topological indices, Earthline Journal of Mathematical Sciences, 2(2) (2019) 343-354.  V.R. Kulli, Multiplicative Kulli-Basava indices, International Journal of Fuzzy Mathematical Archive, 17(1) (2019) 61-67.  V.R. Kulli, Some new topological indices of graphs, International Journal of Mathematical Archive, 10(5) (2019) 62-70.  V.R. Kulli, New Kulli-Basava indices of graphs, International Research Journal of Pure Algebra, 9(7) (2019) 58-63.

 V.R. Kulli, Multiplicative Kulli-Bsava and multiplicative hyper Kulli-Basava indices of some graphs, International Journal of Mathematical Archive, 10(8) (2019) 18-24.

 V.R. Kulli, Multiplicative connectivity KV indices of dendrimers, Journal of Mathematics and Informatics, 15(2019) 1-7.

 V.R. Kulli, Sum connectivity leap index and geometric-arithmetic, leap index of certain windmill graphs, Journal of Global Research in Mathematical Archives, 6(1) (2019) 15-20.

 V.R. Kulli, On connectivity KV indices of certain families of dendrimers, International Journal of Mathematical Archive 10(2) (2019) 14-17.

 V.R.Kulli, B. Chaluvaraju and T.V. Asha, Multiplicative product connectivity and sum connectivity indices of chemical structures in drugs, Research Review International Journal of Multidisciplinary, 4(2) (2019) 949-953.

 V.R. Kulli, Multiplicative connectivity Revan indices of certain families of benzenoid systems, International Journal of Mathematical Archive, 9(3) (2018) 235-241.

 V.R.Kulli, Multiplicative product connectivity and multiplicative sum connectivity indices of dendrimer nanostars, International Journal of Engineering Sciences and Research Technology, 7(2) (2018) 278-283.

 V.R. Kulli, Sum connectivity leap index and geometric-arithmetic, leap index of certain windmill graphs, Journal of Global Research in Mathematical Archives, 6(1) (2019) 15-20.

 V.R. Kulli, B. Chaluvaraju and H.S. Boregowda, Computation of connectivity indices of Kulli path windmill graphs, TWMS Journal of Applied and Engineering Mathematics 8(1a) (2018) 178.

 V.R. Kulli, Computing reduced connectivity indices of certain nanotubes, Journal of Chemistry and Chemical Sciences, 8(11) (2018) 1174-1180.

 V.R. Kulli, On multiplicative F-indices and multiplicative connectivity F-indices of chemical networks, International Journal of Current Research in Science and Technology, 5(2)(2019)1-10.

 V.R. Kulli, Multiplicative Dakshayani indices, International Journal of Engineering Sciences and Research Technology, 7(10) (2018) 75-81.

 V.R. Kulli, Multiplicative KV and multiplicative hyper KV indices of certain dendrimers, International Journal of Fuzzy Mathematical Archive, 17(1) (2019) 13-19.

References

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Aims: To determine quality of life (QoL) outcomes after palliation of pain from bone metastases using magnetic resonance-guided high intensity focused ul- trasound (MR-guided

Contingency samples ( a few milliliters of milk from each donor ) should be frozen be- fore sterilization and filed in the milk bank. “library” for investigational

Continuous input of the drug following floating system administration produces blood drug concentrations within a narrower range compared to the immediate release

Also we determine the multiplicative product connectivity Banhatti index, multiplicative sum connectivity Banhatti index, multiplicative atom bond connectivity Banhatti