New Connectivity Topological Indices

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: [email protected]

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( )=2mr r( −1 .)

Proof: Let G be an r-regular graph with n vertices and m edges. Then Se(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 Cn 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

( )

_n =4 .n

Corollary 1.2. If Kn 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 Wn

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 n2+2n+ +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 Gn

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 Gn 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)+2n 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)+2n 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 Hn

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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