*Vol. 20, No. 1, 2019, 1-8 *

*ISSN: 2279-087X (P), 2279-0888(online) *
*Published on 26 June 2019 *

www.researchmathsci.org

*DOI: http://dx.doi.org/10.22457/apam.622v20n1a1 *

1

Annals of

**New Connectivity Topological Indices **

**V.R.Kulli**Department of Mathematics

Gulbarga University, Gulbarga 585106, India e-mail: vrkulli@gmail.com

*Received 1 June 2019; accepted 25 June 2019 *

* Abstract. New degree based graph indices called Kulli-Basava indices were introduced *
and studied their mathematical and chemical properties which have good response with
mean isomer degeneracy. In this study, we introduce the sum connectivity Kulli-Basava
index, product connectivity Kulli-Basava index, atom bond connectivity Kulli-Basava
index and geometric-arithmetic Kulli-Basava index of a graph and compute exact
formulas for some special graphs.

**Keywords: Connectivity Kulli-Basava indices, wheel, gear, helm graphs **

**AMS Mathematics Subject Classification (2010): 05C05, 05C07, 05C12, 05C35 ****1. Introduction **

*Let G be a finite, simple connected graph with vertex set V(G) and edge set E(G). The *
*degree of dG(v) of a vertex v is the number of vertices adjacent to v. The degree of an *
*edge e = uv in G is defined by dG(e) = dG(u) + dG(v) – 2. The open neighborhood NG(v) of *
*a vertex v is the set of all vertices adjacent to v. The edge neighborhood of a vertex v is *
*the set of all edges incident to v and it is denoted by Ne(v). Let Se(v) denote the sum of the *
*degrees of all edges incident to a vertex v. We refer to [1] for undefined term and *
notation.

A topological index is a numerical parameter mathematically derived from the graph structure. Several topological indices have been considered in Theoretical Chemistry, see [2, 3].

The first and second Kulli-Basava indices were introduced in [4], defined as

( ) ( ) ( )

( )

1 *e* *e* ,

*uv E G*

*KB G* *S u* *S v*

∈

=

## ∑

+ ( ) ( ) ( )

( )

2 *e* *e* .

*uv E G*

*KB G* *S u S v*

∈

=

## ∑

The first and second hyper Kulli-Basava indices were introduced by Kulli [5], defined as

( ) ( ) ( )

( )

2

1 *e* *e* ,

*uv E G*

*HKB G* *S u* *S v*

∈

=

## ∑

+ ( ) ( ) ( )( )

2

2 *e* *e* .

*uv E G*

*HKB G* *S u S v*

∈

2

We introduce the sum connectivity Kulli-Basava index, product connectivity Kulli-Basava index, atom bond connectivity Kulli-Basava index, geometric-arithmetic Kulli-Basava index and reciprocal Kulli-Basava index of a graph, defined as

( )

( ) ( )

( )

1

,

*uv E G*

*e* *e*

*SKB G*

*S* *u* *S* *v*

∈ =

+

## ∑

( )

( ) ( ) ( )

1 ,

*uv E G* _{e}_{e}

*PKB G*

*S* *u S* *v*

∈

=

## ∑

( ) ( ) ( )

( ) ( ) ( )

2 ,

*e* *e*

*uv E G* *e* *e*

*S* *u* *S* *v*
*ABCKB G*

*S* *u S* *v*

∈

+ −

=

## ∑

( ) ( ) ( )

( ) ( )

( )

2

,

*e* *e*

*uv E G* *e* *e*

*S* *u S* *v*
*GAKB G*

*S* *u* *S* *v*

∈ =

+

## ∑

( ) ( ) ( )

( )

.

*e* *e*

*uv E G*

*RKB G* *S* *u S* *v*

∈

=

## ∑

Recently, some connectivity indices were studied [6,7,8,9,10]. In this paper, some connectivity Kulli-Basava indices for some graphs were computed.

**2. Regular graphs **

*A graph G is r-regular 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 *

(i) ( )

( ).

2 1

*m*
*SKB G*

*r r*

=

− (ii) ( ) 2 ( 1).

*m*
*PKB G*

*r r*

= −

(iii) ( ) ( )

( )

4 1 2

.

2 1

*m* *r r*
*ABCKB G*

*r r*

− − =

− (iv) *GAKB G*( )=*m*.

(v) *RKB G*( )=2*mr r*( −1 .)

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

(i) ( )

( ) ( )

( ) ( ) ( ) ( )

1

.

2 1 2 1 2 1

*uv E G*

*e* *e*

*m* *m*

*SKB G*

*S* *u* *S* *v* *r r* *r r* *r r*

∈

= = =

+ − + − −

## ∑

(ii) ( )

( ) ( )

( ) ( ) ( ) ( )

1

.

2 1

2 1 2 1

*uv E G* _{e}_{e}

*m* *m*

*PKB G*

*r r*

*S* *u S* *v* *r r* *r r*

∈

= = =

−

− −

## ∑

(iii) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )

2 2 1 2 1 2

2 1 2 1

*e* *e*

*uv E G* *e* *e*

*S u* *S v* *m* *r r* *r r*

*ABCKB G*

*S u S v* _{r r}_{r r}

∈

+ − − + − −

= =

− −

## ∑

( )

( )

4 1 2

.

2 1

*m* *r r*
*r r*

− − =

3
(iv) ( ) _{( )}( ) ( )_{( )}

( )

( ) ( )

( ) ( )

2 2 2 1 2 1

.

2 1 2 1

*e* *e*

*uv E G* *e* *e*

*S* *u S* *v* *m* *r r* *r r*

*GAKB G* *m*

*S* *u* *S* *v* *r r* *r r*

∈

− −

= = =

+ − + −

## ∑

(v) ( ) ( ) ( )

( )

( ) ( ) ( )

2 1 2 1 2 1 .

*e* *e*

*uv E G*

*RKB G* *S* *u S* *v* *m* *r r* *r r* *mr r*

∈

=

## ∑

= − − = −**Corollary 1.1. If C**n is a cycle with n vertices, then

(i)

### ( )

.2 2

*n*

*n*

*SKB C* = (ii)

### ( )

. 4*n*

*n*
*PKB C* =

(iii)

### ( )

6 .4

*n*

*ABCKB C* = *n* (iv) *GAKB C*

### ( )

*=*

_{n}*n*.

(v) *RKB C*

### ( )

*=4 .*

_{n}*n*

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

(i)

### ( )

1.4 2

*n*

*n n*
*SKB K*

*n*

− =

− (ii)

### ( )

*n*4( 2).

*n*
*SKB K*

*n*

= −

(iii)

### ( )

( )( )( )

4 1 2 2

.

4 2

*n*

*n* *n* *n*

*ABCKB K*

*n*

− − −

=

− (iv)

### ( )

( 1)

. 2

*n*

*n n*

*GAKB K* = −

(v) *RKB K*

### ( )

*=*

_{n}*n n*( −1) (2

*n*−2 .)

**3. Results for wheel graphs **

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

**Figure 1: Wheel W**n

* Lemma 2. Let Wn be a wheel with 2n edges, n*≥3. Then

4

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

(i)

### ( )

2 _{2} _{18}.

2 9
*n*
*n* *n*
*SKB W*
*n*
*n* *n*
= +
+
+ +

(ii)

### ( )

( _{1})( _{9}) 9.

*n*

*n* *n*

*PKB W*

*n*

*n n* *n*

= +

+

+ +

(iii)

### ( )

( )( )

2

2 7 2 16

.

1 9 9

*n*

*n* *n* *n* *n*

*ABCKB W* *n*

*n n* *n* *n*

+ + +

= +

+ + +

(iv)

### ( )

2_{2}( 1)( 9) .

2 9

*n*

*n n n* *n*

*GAKB W* *n*

*n* *n*

+ +

= +

+ +

(v) *RKB W*

### ( )

*n*=

*n n*2+2

*n*+ +9

*n n*( +9 .)

* Proof: Let Wn be a wheel with n+1 vertices and 2n edges. By using definitions and *
Lemma 2, we obtain

(i)

### ( )

( ) ( )

( )

1

*n*

*n*

*uv E W* _{e}_{e}

*SKB W*

*S u* *S v*

∈ =

+

## ∑

( 1) 9 9 9

*n* *n*

*n* *n*

*n n* *n*

= +

+ + + + + +

2 _{2} _{18}.

2 9
*n* *n*
*n*
*n* *n*
= +
+
+ +

(ii)

### ( )

( ) ( )

( )

1

*n*

*n*

*uv E W* _{e}_{e}

*PKB W*

*S u S v*

∈

=

## ∑

( 1)( 9) ( 9)( 9)

*n* *n*

*n n* *n* *n* *n*

= +

+ + + +

( _{1})( _{9}) 9.

*n* *n*

*n*

*n n* *n*

= +

+

+ +

(iii)

### ( )

( ) ( )( ) ( )
( )
2
*n*
*e* *e*
*n*

*uv E W* *e* *e*

*S u* *S v*
*ABCKB W*

*S u S v*

∈

+ −

=

## ∑

( )

( 11)( 99)2 ( 91)( 99)2

*n n* *n* *n* *n*

*n* *n*

*n n* *n* *n* *n*

+ + + − + + + −

= +

+ + + +

( )( ) ( )

2

2 7 2 16

.

1 9 9

*n* *n* *n*

*n* *n*

*n n* *n* *n*

+ + +

= +

+ + +

(iv)

### ( )

( ) ( )( ) ( )
( )
2
*n*
*e* *e*
*n*

*uv E W* *e* *e*

*S u S v*
*GAKB W*

*S u* *S v*

∈ =

+

## ∑

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* *n*

+ +

= +

+ +

(v)

### ( )

( ) ( )( )*n*

*n* *e* *e*

*uv E W*

*RKB W* *S u S v*

∈

=

## ∑

=*n n n*( +1)(

*n*+9)+

*n*(

*n*+9)(

*n*+9)

( )

2

2 9 9 .

*n n* *n* *n n*

5
**4. Results for gear graphs **

*A gear graph is obtained from Wn by adding a vertex between each pair of adjacent rim *
*vertices and it is denoted by Gn. Clearly Gn has 2n+1 vertices and 3n edges. A graph Gn is *
shown in Figure 2.

**Figure 2: A gear graph G**n

* Lemma 4. Let Gn be a gear graph with 2n+1 vertices and 3n edges. Then Gn has two *
types of edges as follows:

*E1={uv *∈* E(Gn)| Se(u) = n (n+1), Se(v) = n+7}, | E1 | = n. *
*E2={uv *∈* E(Gn)| Se(u) = n+7 , Se(v) = 6}, * *| E2 | = 2n. *

**Theorem 5. Let G**n be a gear graph with 2n + 1 vertices 3n edges. Then

(i)

### ( )

2

2 . 13

2 7

*n*

*n* *n*

*SKB G*

*n*
*n* *n*

= +

+

+ +

(ii)

### ( )

( )( ) ( )

2 .

1 9 6 7

*n*

*n* *n*

*PKB G*

*n n* *n* *n*

= +

+ + +

(iii)

### ( )

( )( ) ( )

1 1

2 _{2}

2

2 5 11

2 .

1 7 6 7

*n*

*n* *n* *n*

*ABCKB G* *n* *n*

*n n* *n* *n*

_{+} _{+} _{+}

= +

+ + +

(iv)

### ( )

2_{2}( 1)( 7) 4 6( 7).

2 7 13

*n*

*n n n* *n* *n* *n*
*GAKB G*

*n* *n* *n*

+ + +

= +

+ + +

(v) *RKB G*

### ( )

*n*=

*n n n*( +1)(

*n*+7)+2

*n*6(

*n*+7 .)

* Proof: Let Gn be a gear graph with 2n+1 vertices and 3n edges. By using definitions and *
Lemma 4, we obtain

(i)

### ( )

( ) ( )

( )

1

*n*

*n*

*uv E G*

*e* *e*

*SKB G*

*S u* *S v*

∈ =

+

## ∑

( ) ( )

2 7 6

1 7

*n* *n*

*n*

*n n* *n*

= +

+ + + + +

2

2 . 13

2 7

*n* *n*

*n*
*n* *n*

= +

+

+ +

(ii)

### ( )

( ) ( )

( )

1

*n*

*n*

*uv E G* _{e}_{e}

*PKB G*

*S u S v*

∈

=

## ∑

( )( ) ( )

2

1 7 6 7

*n* *n*

*n n* *n* *n*

= +

6

(iii)

### ( )

( ) ( )( ) ( )

( )

2

*n*

*e* *e*

*n*

*uv E G* *e* *e*

*S u* *S v*
*ABCKB G*

*S u S v*

∈

+ −

=

## ∑

( ) ( )

( )( ) ( )

1 1

2 2

1 7 2 7 6 2

2

1 7 7 6

*n n* *n* *n*

*n* *n*

*n n* *n* *n*

_{+ + + −} _{+ + −}

= +

+ + +

( )( ) ( )

1 1

2 _{2}

2

2 5 11

2 .

1 7 6 7

*n* *n* *n*

*n* *n*

*n n* *n* *n*

_{+} _{+} _{+}

= +

+ + +

(iv)

### ( )

( ) ( )( ) ( )

( )

2

*n*

*e* *e*

*n*

*uv E G* *e* *e*

*S u S v*
*GAKB G*

*S u* *S v*

∈ =

+

## ∑

2_{(}(

_{)}1)( 7) 2 2 ( 7 6)

1 7 7 6

*n* *n n* *n* *n* *n*

*n n* *n* *n*

+ + +

= +

+ + + + +

( )( ) ( )

2

2 1 7 4 6 7

2 7 13

*n n n* *n* *n* *n*

*n* *n* *n*

+ + +

= +

+ + +

(v)

### ( )

( ) ( )( )*n*

*n* *e* *e*

*uv E G*

*RKB G* *S u S v*

∈

=

## ∑

=*n n n*( +1)(

*n*+7)+2

*n*6(

*n*+7 .)

( )

2

2 9 2 6 7 .

*n n* *n* *n* *n*

= + + + +

**5. Results for helm graphs **

A helm graph *Hn is a graph obtained from Wn by attaching an end edge to each rim *
vertex. Clearly Hn has 2n+1 vertices and 3n edges. A graph Hn is depicted in Figure 3.

**Figure 3: A graph H**n

* Lemma 6. If Hn is a helm graph with 2n+1 vertices and 3n edges, then Hn has three types *
of edges as

*E1={uv *∈ E(Hn)| Se(u) = n(n+2), Se(v) = n + 17} | E1 | = n.
*E2={uv *∈ E(Hn)| Se(u) = Se(v) = n + 17}, | E2 | = n.
*E3*={uv ∈ E(Hn)| Se(u) = n +17, Se(v) = 3} | E3 | = n.

* Theorem 7. If Hn* is a helm graph with 2n+1 vertices and 3n edges, then

(i)

### ( )

2 _{2} _{34} _{20}.

3 17

*n*

*n* *n* *n*

*SKB H*

*n* *n*
*n* *n*

= + +

+ +

7

(ii)

### ( )

( _{2})( _{17}) ( 17) _{3}( _{17}).

*n*

*n* *n* *n*

*PKB H*

*n*

*n n* *n* *n*

= + +

+

+ + +

(iii)

### ( )

( )( ) ( )

1 1

2 _{2}

2

3 5 2 32 18

.

2 17 17 3 17

*n*

*n* *n* *n* *n*

*ABCKB H* *n* *n* *n*

*n n* *n* *n* *n*

_{+} _{+} _{+} _{+}

= + +

+ + + +

(iv)

### ( )

2_{2}( 1)( 7) 2 3( 17).

3 7 20

*n*

*n n n* *n* *n* *n*

*GAKB H* *n*

*n* *n* *n*

+ + +

= + +

+ + +

(v) *RKB H*

### ( )

*=*

_{n}*n n n*( +2)(

*n*+17)+

*n n*( +17)+

*n*2(

*n*+17 .)

* Proof: Let Hn be a helm graph with 2n+1 vertices and 3n edges. Then by using *
definitions and Lemma 6, we deduce

(i)

### ( )

( ) ( )

( )

1

*n*

*n*

*uv E H* _{e}_{e}

*SKB H*

*S u* *S v*

∈ =

+

## ∑

( )

2

17 17 17 3

2 17

*n* *n* *n*

*n* *n* *n*

*n n* *n*

= + +

+ + + + +

+ + +

2 _{2} _{34} _{20}.

3 17

*n* *n* *n*

*n* *n*
*n* *n*

= + +

+ +

+ +

(ii)

### ( )

( ) ( )

( )

1

*n*

*n*

*uv E H* _{e}_{e}

*PKB H*

*S u S v*

∈

=

## ∑

( 2)( 17) ( 17)( 17) ( 17 3)

*n* *n* *n*

*n n* *n* *n* *n* *n*

= + +

+ + + + +

( _{2})( _{17}) ( 17) _{3}( _{17}).

*n* *n* *n*

*n*

*n n* *n* *n*

= + +

+

+ + +

(iii)

### ( )

( ) ( )( ) ( )

( )

2

*n*

*e* *e*

*n*

*uv E H* *e* *e*

*S u* *S v*
*ABCKB H*

*S u S v*

∈

+ −

=

## ∑

( )

( )( ) ( )( )

1 2

2 17 2 17 17 2

2 17 17 17

*n n* *n* *n* *n*

*E* *E*

*n n* *n* *n* *n*

+ + + − + + + −

= +

+ + + +

( )

3

17 3 2

17 3

*n* *n*

*E*

*n*

+ + + − +

+

( )( ) ( )

1 1

2 _{2}

2

3 15 2 32 18

.

2 17 17 3 7

*n* *n* *n* *n*

*n* *n* *n*

*n n* *n* *n* *n*

_{+} _{+} _{+} _{+}

= + +

+ + + +

(iv)

### ( )

( ) ( )( ) ( )

( )

2

*n*

*e* *e*

*n*

*uv E H* *e* *e*

*S u S v*
*GAKB H*

*S u* *S v*

∈ =

+

## ∑

( )( )

( )

( )( )

1 2

2 2 17 2 17 17

2 17 17 17

*n n* *n* *n* *n*

*E* *E*

*n n* *n* *n* *n*

+ + + +

= +

8

( )

3

2 17 3

17 3

*n*
*E*

*n*

+ +

+ +

( )( ) ( )

2

2 2 7 2 3 17

. 20 3 17

*n n n* *n* *n* *n*

*n*
*n*

*n* *n*

+ + +

= + +

+

+ +

(v)

### ( )

( ) ( )( )*n*

*n* *e* *e*

*uv E H*

*RKB H* *S u S v*

∈

=

## ∑

( )( ) ( )( ) ( )

1 2 17 2 17 17 3 17 3.

*E* *n n* *n* *E* *n* *n* *E* *n*

= + + + + + + +

( 2)( 17) ( 17) 3( 17 .)

*n n n* *n* *n n* *n* *n*

= + + + + + +

**Acknowledgement. The author is thankful to the referee for his/her suggestions. ****REFERENCES **

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India (2012).

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

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