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ISSN 2229 – 5046

SOME NEW TOPOLOGICAL INDICES OF GRAPHS

V. R. KULLI*

Department of Mathematics,

Gulbarga University, Gulbarga 585106, India.

(Received On: 15-03-19; Revised & Accepted On: 24-04-19)

ABSTRACT

I

n 2019, Basavanagoud et al. introduced new degree based topological indices called Kulli-Basava indices and studied their mathematical and chemical properties which have good response with mean isomer degeneracy. In this paper, we propose the first and second hyper Kulli-Basava indices and their polynomials of a graph and compute exact formulae for complete graphs, wheel graphs, gear graphs and helm graphs. Also we determine the Kulli-Basava indices of gear graphs and helm graphs.

Key words: hyper Kulli-Basava indices, polynomials, graphs.

Mathematics Subject Classification: 05C07, 05C76, 92E10.

1. INTRODUCTION

In Chemical Sciences, topological indices have been found to be useful in chemical documentation, isomer discrimination, structure property relationships, structure activity relationships. There has been considerable interest in the general problem of determining topological indices.

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 edges incident 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. The set of all vertices adjacent to v is called the open

neighborhood of v and denoted by NG(v). Let SG(v) denote the sum of the degrees of all vertices adjacent to a vertex v.

The set of all edges incident to v is called the edge neighborhood of v and 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.

The first and second Zagreb indices are the degree based topological indices, introduced by Gutman and Trinajstić in [2]. These indices have many applications in Chemistry. Recently, Basavanagoud and Jakkannavar introduced [3] the following Kulli-Basava indices:

The first Kulli-Basava index of a graph G is defined as

     

 

1 e e .

uv E G

KB G S u S v

 

The modified first Kulli-Basava index of a graph G is defined as

   

 

2 *

1 e .

u V G

KB G S u

The second Kulli-Basava index of a graph G is defined as

     

 

2 e e .

uv E G

KB G S u S v

The third Kulli-Basava index of a graph G is defined as

     

 

3 e e .

uv E G

KB G S u S v

Corresponding Author: V. R. Kulli*

(2)

Considering the above Kulli-Basava indices, we define the following polynomials:

The first Kulli-Basava polynomial of a graph G is defined as

     

1 , .

e e

S u S v

uv E G

KB G x x  

The modified first Kulli-Basava polynomial of a graph G is defined as

 

 

2

*

1 , .

e

S u

u V G

KB G x x

The second Kulli-Basava polynomial of a graph G is defined as

   

 

2 , .

e e

S u S v

uv E G

KB G x x

The third Kulli-Basava polynomial of a graph G is defined as

     

3 , .

e e

S u S v

uv E G

KB G x x

In a recent years, the Zagreb index [4], F-index [5], sum connectivity index [6], reverse indices [7], Revan indices [8], Dakshayani index [9] were introduced and extensively studied.

The hyper Zagreb index of a graph G was introduced by Shirdel et al. in [10] and it is defined as

     

 

2

1 G G .

uv E G

HM G d u d v

 

In [11], Gao et al. proposed the second hyper Zagreb index, defined as

     

 

2

2 G G .

uv E G

HM G d u d v

 

We now introduce the first and second hyper Kulli-Basava indices of a graph G, 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

 

Considering the first and second hyper Kulli-Basava indices, we propose the first and second hyper Kulli-Basava polynomials of a graph, defined as

     

2

1 , ,

e e

S u S v

uv E G

HKB G x x  

   

 

2

2 , .

e e

S u S v

uv E G

HKB G x x 

2. COMPLETE GRAPHS

Let Kn be a complete graph with n vertices. Then the degree of each vertex of Kn is n – 1 and the number of edges in Kn

is  1. 2 n n

Lemma 1: Let Kn be a complete graph with n vertices, n≥ 2. Then

(i) V1 = {u V(Kn) | Se(u) = 2(n – 1)(n – 2)}, |V1| = n.

(ii) E1 = {uv V(Kn) | Se(u) = Se(v) = 2(n – 1)(n – 2)}, |E1| =

 1

. 2 n n

Theorem 2: If Kn is a complete graph with n≥ 2 vertices, then

  4 1 2

1

1 ,

2

n n n

n n

(3)

  2 2

* 4 1 2

1 ,

n n

n

KB K xnx  

    2 2

4 1 2

2 1 , 2 n n n n n

KB K x   x  

  0

3 1 , . 2 n n n

KB K x   x

Proof: By using definitions and Lemma 1, we obtain

   

 

       

4 1 2 4 1 2

1 1 1 , 2 e e n

S u S v n n n n

n

uv E K

n n

KB K x xE x   x  

  

 

           

2 2 2 2 2

* 4 1 2 4 1 2

1 , 1

e

n

S u n n n n

n

u V K

KB K x x V x   nx  

 

   

 

  2 2     2 2

4 1 2 4 1 2

2 1 1 , 2 e e n

S u S v n n n n

n

u E K

n n

KB K x x E x   x  

  

 

        0 0 3 1 1 , . 2 e e n

S u S v n

uv E K

n n

KB K x xE x x

 

Theorem 3: The first hyper Kulli-Basava index and its polynomial of a complete graph are given by (i) HKB K1

 

n 8n n 1 3 n22

(ii)

     

2 2

16 1 2 1 1 , . 2 n n n n n

HKB K x   x  

Proof:

(i) Let Kn be a complete graph with n≥ 2 vertices. By using definition and Lemma 1, we derive

 

   

 

  

2 2

1 1 4 1 2

n

n e e

uv E K

HKB K S u S v E n n

 

  

  3 2

8n n 1 n 2 .

  

(ii) From definition and by using Lemma 1, we obtain

   

 

  

 

2 2

4 1 2

1 , 1

e e

n

S u S v n n

n

uv E K

HKB K x x   E x  

    2 2

16 1 2 1

. 2

n n

n n

x  

 

Theorem 4: The second hyper Kulli-Basava index and its polynomial of a complete graph are given by (i) HKB K2

 

n 8n n 1 5 n2 .4

(ii)

     

4 4

16 1 2 2 1 , . 2 n n n n n

HKB K x   x  

Proof:

(i) Let Kn be a complete graph with n≥ 2 vertices. Then by using definition and Lemma 1, we deduce

 

   

 

    2

2 2 2

2 1 4 1 2

n

n e e

uv E K

HKB K S u S v E n n

 

 

 

  5 4

8n n 1 n 2 .

  

(ii) By using definition and Lemma 1, we obtain

   

 

    2

2 2 2

4 1 2

2 , 1

e e

n

S u S v n n

n

uv E K

HKB K x x  E x    

    4 4

16 1 2 1

. 2

n n

n n

x  

(4)

3. WHEEL GRAPHS

A wheel Wn is the join of K1 and Cn. Clearly Wn has n+1vertices and 2n edges. A graph Wn is depicted in Figure 1. The

vertices of Cn are called rim vertices and the vertex of K1 is called apex.

Figure-1: Wheel Wn

Lemma 5: Let Wn be a wheel with n+1 vertices. Then Wn has two types of vertices as given below:

V1 = {uV(Wn) | Se(u) = n(n+1)}, |V1| = 1.

V2 = {uV(Wn) | Se(u) = n+9}, |V2| = n.

Lemma 6: Let Wn be a wheel with 2n edges. Then Wn has two types of edges as given below:

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

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

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

2

2 9 2 18

1 ,

n n n

n

KB W xnx   nx

2 2  2

* 1 9

1 ,

n n n

n

KB W xx  nx

    2

1 9 9

2 ,

n n n n

n

KB W xnx   nx

2

9 0

3 ,

n n

KB W xnx  nx

Proof: By using definitions and Lemmas 5 and 6, we deduce

   

 

 

9 1 9 9

1 , 1 2

e e

n

S u S v n n n n n

n

uv E W

KB W x xE x    E x   

 

2

2 9 2 18

n n n

nx   nx

 

 

 

   

2 2 2 2

* 1 9

1 , 1 2

e

n

S u n n n

n

u V W

KB W x x V xV x

 

 2  2

2 1 9

n n n

xnx

 

   

 

 9  1  9 9

2 , 1 2

e e

n

S u S v n n n n n

n

uv E W

KB W x x E x   E x  

 

    2

1 9 9

n n n n

nx   nx

 

   

 

 

9 1 9 9

3 , 1 2

e e

n

S u S v n n n n n

n

uv E W

KB W x xE x    E x   

 

2

9 0

n

nxnx

 

Theorem 8: The first hyper Kulli-Basava index and its polynomial of a wheel Wn are given by

(i) HKB W1

 

nn n

22n9

2n2n18 .2

(ii)

   

2 2

2

2 9 2 18

1 , .

n n n

n

(5)

Proof:

(i) Let Wn be a wheel with n≥ 3 vertices. Then by using definition and Lemma 6, we deduce

 

   

 

 

 

2 2 2

1 1 9 1 2 9 9

n

n e e

uv E W

HKB W S u S v E n n n E n n

 

       

2

2  2

2 9 2 18 .

n n n n n

    

(ii)

   

 

2

1 ,

e e

n

S u S v n

uv E W

HKB W x x  

   

2 2

2

2 9 2 18

.

n n n

nx   nx

 

Theorem 9: The second hyper Kulli-Basava index and its polynomial of a wheel Wn are given by

(i) HKB W2

 

nn n 92n2n12n92.

(ii)

     

2 2 4

2 1 9 9

2 , .

n n n n

n

HKB W xnx   nx

Proof:

(i) By using definition and Lemma 6, we derive

 

   

 

   

  

2 2 2

2 1 9 1 2 9 9

n

n e e

uv E W

HKB W S u S v E n n n E n n

 

     

 2 2 2  2

9 1 9 .

n nn n n

    

(ii)

   

 

2

2 ,

e e

n

S u S v n

uv E W

HKB W x x 

     

2 2 4

2

1 9 9

.

n n n n

nx   nx

 

4. GEAR GRAPHS

A graph is a gear graph 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: Gear graph Gn

Lemma 10: Let Gn be a gear graph with 2n+1 vertices. Then Gn has three types of vertices as given below.

V1 = {uV(Gn) | Se(u) = n(n+1)}, | V1 | = 1.

V2 = {uV(Gn) | Se(u) = n+7)}, | V2 | = n.

V3 = {uV(Gn) | Se(u) = 6)}, | V3 | = n.

Lemma 11: Let Gn be a gear graph with 3n edges. Then Gn has two types of edges as shown below.

E1 = {uvE(Gn) | Se(u) = n(n+1), Se(v) = n+7}, | E1 | = n.

E2 = {uvE(Gn) | Se(u) = n+7, Se(v) = 6}, | E2 | = 2n.

Theorem 12: Let Gn be a gear graph with 2n+1 vertices and 3n edges. Then

 

3 2

1 n 4 33 .

KB Gnnn

 

* 4 3 2

1 n 3 15 85 .

(6)

 

 

2

2 n 7 12 .

KB Gn nn  n

 

2  

3 n | 7 | 2 1 .

KB Gn n   n n

Proof: By using definitions and Lemmas 10 and 11, we deduce

 

   

  1

n

n e e

uv E G

KB G S u S v

 

 

1 7

2  7 6

n n n n n n

      

3 2

4 33 .

n n n

  

 

 

  2 *

1

n

n e

u V G

KB G S u

 

2  2 2

1 7 6

n n n n n

    

4 3 2

3 15 85 .

n n n n

   

 

   

  2

n

n e e

uv E G

KB G S u S v

  

1 7

2

 7 6

n n n n n n

    

7

2 12 .

n n n n

   

 

   

  3

n

n e e

uv E G

KB G S u S v

 1  7 2 7 6

n n n n n n

      

 

2 7 2 1 .

n n n n

   

Theorem 13: Let Gn be a gear graph with 2n+1 vertices and 3n edges. Then

2

2 7 13

1 , 2 .

n n n

n

KB G xnx    nx

2 2  2

* 1 7 36

1 , .

n n n

n

KB G xx  nx  nx

 1 7 6 7

2 , 2 .

n n n n

n

KB G xnx    nx

2 7 1

3 , 2 .

n n

n

KB G xnx   nx

Proof: By using definitions and Lemmas 10 and 11, we derive

   

 

 1 7 7 6

1 , 1 2

e e

n

S u S v n n n n

n

uv E G

KB G x xE x    E x  

 

2

2 7 2 13

n n n

nx   nx

 

 

 

   

2 2 2 2 2

* 1 7 6

1 , 1 2 3

e

n

S u n n n

n

u V G

KB G x x V xV xV x

  

 2  2 2

1 7 36.

n n n

xnxnx

  

   

 

 1 7 6 7

2 , 1 2

e e

n

S u S v n n n n

n

uv E G

KB G x x E x   E x

 

 1 7 6 7 2

n n n n

nx   nx

 

   

 

 1  7 7 6

3 , 1 2

e e

n

S u S v n n n n

n

uv E G

KB G x xE x    E x  

 

2 7 1

2 .

n n

nxnx

 

Theorem 14: The first hyper Kulli-Basava index and its polynomial of a gear graph Gn are given by

(i) HKB G1

 

nn n

22n7

2 2n n 13 .2

(ii)

   

2 2

2 2 7 13

1 , 2 .

n n n

n

(7)

Proof: Let Gn be a gear graph with 2n+1 vertices and 3n edges.

(i) By using definition and Lemma 11, we obtain

 

   

 

 

 

2 2 2

1 1 1 7 2 7 6

n

n e e

uv E G

HKB G S u S v E n n n E n

 

      

2

2  2

2 7 2 13 .

n n n n n

    

(ii)

   

 

2

1 ,

e e

n

S u S v n

uv E G

HKB G x x  

   

2 2

2

2 7 2 13 .

n n n

nx   nx

 

Theorem 15: The second hyper Kulli-Basava index and its polynomial of a gear graph Gn are given by

(i) HKB G2

 

nn n 72n2n1272 .

(ii)

     

2 2 2

2

1 7 36 7

2 , 2 .

n n n n

n

HKB G xnx    nx

Proof:

(i) By using definition and Lemma 11, we have

 

   

 

  

 

2 2 2

2 1 1 7 2 7 6

n

n e e

uv E G

HKB G S u S v E n n n E n

 

    

 2 2 2

7 1 72 .

n nn n

   

(ii)

   

 

2

2 ,

e e

n

S u S v n

uv E G

HKB G x x 

     

2 2 2

2

1 7 2 36 7 .

n n n n

nx   nx

 

5. 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, see Figure 3.

Figure-3: Helm graph Hn

Lemma 16: Let Hn be a helm graph with 2n+1 vertices.

Then Hn has three types of vertices as follows:

V1 = {uV(Hn) | Se(u) = n(n+2)}, | V1 | = 1.

V2 = {uV(Hn) | Se(u) = n+17}, | V2 | = n.

V3 = {uV(Hn) | Se(u) = 3}, | V3 | = n.

Lemma 17: Let Hn be a helm graph with 3n edges. Then Hn has three types of edges as follows.

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.

(8)

Theorem 18: Let Hn be a helm graph with 2n + 1 vertices and 3n edges. Then

 

3 2

1 n 3 4 71 .

KB Hnnn

 

* 4 3 2

1 n 5 38 298 .

KB Hnnnn

 

 

2

2 n 7 3 20 .

KB Hn nnn

 

2  

3 n 17 14 .

KB Hn n  nn n

Proof: By using definitions and Lemmas16 and 17,we obtain

 

   

  1

n

n e e

uv E H

KB H S u S v

 

 

2 17

 17 17  17 3

n n n n n n n n n

          

3 2

3n 4n 71 .n

  

 

 

  2 *

1

n

n e

u V H

KB H S u

 

2  2 2

2 17 3

n n n n n

     

4 5 3 38 2 298 .

n n n n

   

 

   

  2

n

n e e

uv E H

KB H S u S v

  

2 17

 17 17

 17 3

n n n n n n n n n

       

 

2

17 3 20 .

n n n n

   

 

   

  3

n

n e e

uv E H

KB H S u S v

 2  17 | 17  17 | 17 3

n n n n n n n n n

          

 

2

17 14 .

n n n n n

    

Theorem 19: Let Hn be a helm graph with 2n+1 vertices and 3n edges. Then

2 3 17 21720

1 , .

n n n n

n

KB H xnx   nx  nx

 2  2

* 2 17 9

1 , .

n n n

n

KB H xx  nx  nx

    2  

2 17 17 3 17

2 , .

n n n n n

n

KB H xnx   nx  nx

2 17 0 14

3 , .

n n n

n

KB H xnx   nxnx

Proof: By using definitions and Lemmas 16 and 17, we deduce

   

 

1 ,

e e

n

S u S v n

uv E H

KB H x x

 2 17 17 17 17 3

.

n n n n n n

nx    nx    nx 

  

 

2

3 17 2 17 20.

n n n n

nx   nxnx

  

 

 

2

*

1 ,

e

n

S u n

u V H

KB H x x

 

 2  2

2 17 9

.

n n n

xnxnx

  

   

 

2 ,

e e

n

S u S v n

uv E H

KB H x x

    2  

2 17 17 3 17

.

n n n n n

nx   nxnx

  

   

 

3 ,

e e

n

S u S v n

uv E H

KB H x x

 2  17  17  17 17 3 .

n n n n n n

nx    nx    nx  

  

2 17 0 14

.

n n n

nx   nx nx

(9)

Theorem 20: The first hyper Kulli-Basava index and its polynomial of a helm graph Hn are

(i) HKB H1

 

nn n

23n17

2 4n n 172 n n 20 .2

(ii)

     

2 2 2

2 3 17 2 34 20

1 , .

n n n n

n

HKB H xnx   nx  nx

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

(i) By using definition and Lemma 17, we deduce

 

   

 

 

   

2 2 2 2

1 2 17 17 17 17 3

n

n e e

uv E H

HKB H S u S v n n n n n n n n n

 

          

n n

23n17

2 4n n 172n n 20 .2

(ii)

   

 

2

1 ,

e e

n

S u S v n

uv E H

HKB H x x  

     

2 2 2

2 3 7 2 34 20

.

n n n n

nx   nxnx

  

Theorem 21: The second hyper Kulli-Basava index or its polynomial of a helm graph Hn are

(i) HKB2

 

Hnn n 172n2n22 n1729 .

(ii)

       

2 2 4 2

2

2 17 17 9 17

2 , 2 .

n n n n n

n

HKB H xnx   nx   nx

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

(i)

 

   

 

  

  

 

2 2 2 2

2 2 17 17 17 17 3

n

n e e

uv E H

HKB H S u S v n n n n n n n n n

 

       

 2 2 2  2

7 2 17 9 .

n nn n n

     

(ii)

   

 

2

2 ,

e e

n

S u S v n

uv E H

HKB H x x 

      

2 4 2

2 17 17 9 17

.

n n n n n

nx   nxnx

  

REFERENCES

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

2. I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, (1972) 535-538.

3. B. Basavanagoud, P. Jakkannavar, Kulli-Basava indices of graphs, Inter. J. Appl. Eng. Research, 14(1) (2018) 325-342.

4. G.H. Fath-Tabar, Old and new Zagreb indices of graphs, MATCH Common. Math. Comput. Chem. 65 (2011) 79-84.

5. B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015), 1184-1190. 6. B. Zhou and N. Trinajstic, On a novel connectivity index, J. Math. Chem. 46(2009) 1252-1270.

7. S. Ediz, Maximal graphs of the first reverse Zagreb beta index, TWMS J.Appl.Eng. Math. 8 (2018) 306-310. 8. V.R. Kulli, Revan indices of oxide and honeycomb networks, International Journal of Mathematics and its

Applications, 5(4-E) (2017) 663-667.

9. V.R. Kulli, Dakshayani indices, Annals of Pure and Applied Mathematics, 18(2) (2018) 139-146.

10. G.H. Shirdel, H. Rezapour and A.M. Sayadi, The hyper-Zagreb index of graph operations, Iranian J. Math. Chem. 4(2) (2013) 213-220.

11. W. Gao, M.R. Farahani, M. K. Siddiqui, M. K. Jamil, On the first and second Zagreb and first and second hyper Zagreb indices of carbon nanocones CNCk[n], J. Comput. Theor. Nanosci. 13 (2016) 7475-7482.

Source of support: Nil, Conflict of interest: None Declared.

References

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