• No results found

The First and Second Zagreb Indices for Some Special Graphs SK International Journal of Multidisciplinary Research Hub

N/A
N/A
Protected

Academic year: 2022

Share "The First and Second Zagreb Indices for Some Special Graphs SK International Journal of Multidisciplinary Research Hub"

Copied!
7
0
0

Loading.... (view fulltext now)

Full text

(1)

ISSN: 2394-3122 (Online) Volume 2, Issue 10, October 2015

SK International Journal of Multidisciplinary Research Hub

Journal for all Subjects

Research Article / Survey Paper / Case Study Published By: SK Publisher (www.skpublisher.com)

The First and Second Zagreb Indices for Some Special Graphs

U. Mary1

Department of Mathematics Nirmala College for Women Coimbatore, Tamil Nadu, India

Anju Antony2

Research scholar in Mathematics Nirmala College for Women Coimbatore, Tamil Nadu, India

Abstract:A topological index is a map from the set of chemical compounds represented by molecular graphs to the set of real numbers. Let be a simple graph. The first Zagreb index is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph In this paper, the First and Second Zagreb indices of Friendship graph, Wheel graph and Barbell graph [8] are obtained.

Keywords: PyrimidineBarbell graph, First Zagreb Index, Friendship graph, Second Zagreb Index, Wheel graph.

I. INTRODUCTION

Long time ago [4], within the study of dependence of total

electron energy on molecular structure, some expressions were deduced, containing the following terms 1

 ( )

2

vertices

d

i

M

and j

edges

i d

d

M2

with

d

i stands for the degree (number

of first neighbours) of the vertex

v

i of the molecular graph. These terms are in fact measures of branching of the molecular carbon – atom skeleton [3] and can thus be viewed as molecular structure- descriptors [1]. In the chemical literature, M1and M2are called the first Zagreb Group Index and the Second Zagreb Group index respectively, or in short the first Zagreb Index and the Second Zagreb Index.

Independently of its chemical context, the sum of squares of vertex degrees of a graph was studied by quite a few mathematicians; for details and references see [7]. As a consequence, numerous results on M1are known.

II. THE FIRST ZAGREB INDEX FOR SOME SPECIAL TYPES OF GRAPHS

2.1 Wheel Graph

A wheel graph is a graph with

p

vertices, formed by connecting a single vertex to all vertices of

( p  1 )

cycle. It is denoted as

W

p1or

W

p

.

Wheel graphs are planar graphs and as such have a unique planar embedding. They are self-dual, the planar dual of any wheel graph is an isometric graph. Any maximal planar graph other thanK4W4, contain as a sub graph either

W

5 or

W

6

.

There is always a Hamiltonian cycle in the wheel graph and there are

( p

2

 3 p  3 )

cycles in

W

p[6].

2.2 Theorem: Let

W

pbe the wheel graph of order

p

where p4 and the First Zagreb index of the wheel graph Wpis

(2)

Volume 2, Issue 10, October 2015 pg. 1-7

).

8 )(

1 ( )

1

( W p p

M

p

  

Proof:

The First Zagreb index of

W

p for p4 can be computed as follows.

By the definition of first Zagreb index, 1

 (deg )

2

.

V v

v M

ForW4, 3 vertices of W4 have degree 3 and 1 vertex of W4 have degree 3 For

W

5, 4 vertices of

W

5 have degree 3 and 1 vertex of

W

5 have degree 4

For

W

6, 5 vertices of

W

6 have degree 3 and 1 vertex of

W

6 have degree 5 We can observe that

36 ) (

4

1

W

M

M

1

( W

5

)  52

M

1

( W

6

)  70

M

1

( W

7

)  90

For a Wheel graph

W

p

,

the outer vertex has degree 3 and the center vertex have degree

( p  1 ).

Therefore, the first Zagreb index of the wheel graph

W

p is

2

1

( )  (deg )

V v

p

v

W M

) 8 )(

1 (

) 1 .(

1 ) 3 )(

1

( 2 2

p p

p p

Thus, First Zagreb Index of the wheel graph Wpis

M

1

( W

p

)  ( p  1 )( 8  p ) for p  4 .

2.3. Barbell Graph

A

p

- Barbell graph is the simple graph obtained by connecting two copies of a complete graph Kp by a bridge and it is denoted by Bp[9].

(3)

Volume 2, Issue 10, October 2015 pg. 1-7

2.4 Theorem: Let

B

pbe the Barbell graph of order

p

, where (p3)and the First Zagreb index of the Barbell graph is

)

1(Bp

M

 2 [( p  1 )

3

p

2

].

Proof:

The First Zagreb index of Bp for p3 can be computed as follows.

By the definition of first Zagreb index, 1

 (deg )

2

.

V v

v M

For

B

3, 2*2 vertices of

B

3 have degree 2 and 2 vertices of

B

3have degree 3 ForB4, 2*3 vertices of B4 have degree 3 and 2 vertices of B4 have degree 4 For

B

5, 2*4 vertices of

B

5 have degree 4 and 2 vertices of

B

5 have degree 5

For a Barbell graphBp, 2(p1)vertices of Bphave degree (p1) and 2 vertices of Bphave degree

p

.

The First Zagreb Index of

B

3, M1(B3)4(2)2 2(3)2 34. We observe that,

86 ) ( 4

1 B

M

M

1

( B

5

)  178 M

1

( B

6

)  322

and so on.

1

   (deg )

2

V v

p

v

B

M

 2 ( p  1 )( p  1 )

2

 2 . p

2

] )

1 [(

2 p

3

p

2

Thus, the First Zagreb index of the Barbell graph is M1(Bp)

 2 [( p  1 )

3

p

2

].

2.5 Friendship Graph

The friendship graph

F

p can be constructed by joining

p

copies of the cycle graph with a common vertex and it is denoted as

F

p

. F

p is a planar undirected graph with

( 2 p  1 )

vertices and

3 p

edges.

(4)

Volume 2, Issue 10, October 2015 pg. 1-7

In this graph, every two vertices have exactly one neighbour in common. If a group of people has the property that every pair of people has exactly a friend in common, then there must be one person who is a friend to all the others[5].

2.6 Theorem: Let

F

p be the Friendship graph of order

p

where p2and the First Zagreb index of the Friendship graphis

) 2 ( 4 )

1

( F p p

M

p

 

. Proof:

The First Zagreb index of

W

p for p2 can be computed as follows.

By the definition of first Zagreb index, 1

 (deg )

2

V v

v M

ForF2, 4 vertices of F2 have degree 2 and 1 center vertex of F2 have degree 4 For

F

3, 6 vertices of

F

3 have degree 2 and 1 center vertex of

F

3 have degree 6

For

F

4, 8 vertices of

F

4 have degree 2 and 1 center vertex of

F

4 have degree 8 For

F

5, 10 vertices of

F

5 have degree 2 and 1 center vertex of

F

5 have degree 10

For a Friendship graph

F

p

, 2 p

vertices have degree 2 and 1 center vertex have degree

2 p .

The First Zagreb Index of

F

2, M1(F2)4.2(22)32 We observe that,

32 ) (

2

1

F

M

M

1

( F

3

)  60

M

1

( F

4

)  96

M

1

( F

5

)  140

The first Zagreb index of Friendship graph

F

p

,

1

( )  (deg )

2

V v

p

v

F M

) 2 ( 4

) 2 4 ( 2

) 2 ( ) 2 (

2

2 2

p p

p p

p p

Thus,

M

1

( F

p

)  4 p ( 2  p )

for

p  2 .

(5)

Volume 2, Issue 10, October 2015 pg. 1-7 III. THE SECOND ZAGREB INDEX FOR SOME SPECIAL TYPES OF GRAPHS

The second Zagreb index for Wheel graph, Barbell graph and Friendship graph are computed in this section.

3.1 Theorem: Let Wp be the wheel graph of order

p

where

p  4

and the Second Zagreb index of the wheel graph Wpis

) 2 )(

1 ( 3 )

2

( W p p

M

p

  

Proof:

The Second Zagreb index of

W

p for p4 can be computed as follows.

By the definition of Second Zagreb index, j

edges

i d

d

M2

where di deg(vi),dj deg(vj)and

v

i &vj are adjacent

vertices.

The wheel graph W4 has 3 edges with end vertices of degree 3 and 3 and 3 edges with end vertices of degree 3 and 3.

The wheel graph

W

5 has 4 edges with end vertices of degree 3 and 3 and 4 edges with end vertices of degree 3 and 4.

The wheel graph

W

6 has 5 edges with end vertices of degree 3 and 3 and 5 edges with end vertices of degree 3 and 5.

Therefore, for any

p

, the wheel graph Wp has

( p  1 )

edges with end vertices of degree 3 and 3 and

( p  1 )

edges with end vertices of degree 3 and

( p  1 )

.

We observe that, 54 ) ( 4

2 W

M

M

2

( W

5

)  84

M

2

( W

6

)  120

M

2

( W

7

)  162

The Second Zagreb index of Wp is given by

) 2 )(

1 ( 3

) 3 3 )(

1 ( ) 1 ( 9

)]

1 .(

3 )[

1 ( ) 3 . 3 )(

1 ( )

2

(

p p

p p

p

p p

p W

M

P

Hence,

M

2

( W

p

)  3 ( p  1 )( 2  p )

for

p  4 .

3.2 Theorem: Let Bpbe the Barbell graph of order

p

, where (p3)and the Second Zagreb index of the barbell graph is

) . 1 )(

2 2 ( . . 1 )]

1 )(

1 )[(

2 3 ( )

(

2

2

B p p p p p p p p p

M

p

        

Proof:

By the definition of Second Zagreb index, j

edges

i d

d M2

The Barbell graph

B

3 has 4 edges with end vertices of degree 2 and 2, the center edge (bridge) of

B

3 has 1 edge with end vertices of degree 3 and 3, and 4 edges with end vertices of degree 2 and 3.

The Barbell graph B4 has 6 edges with end vertices of degree 3 and 3, the center edge (bridge) of

B

3 has 1 edge with end

(6)

Volume 2, Issue 10, October 2015 pg. 1-7 The Barbell graph

B

5 has 12 edges with end vertices of degree 4 and 4, the center edge (bridge) of

B

5 has 1 edge with end vertices of degree 5 and 5, and 8 edges with end vertices of degree 3 and 4.

Therefore the Barbell graph Bp has

( p

2

 3 p  2 )

edges with end vertices of degree

( p  1 )

and

( p  1 ),

the center edge (bridge) of Bphas 1 edge with end vertices of degree

p

and

p

and

( 2 p  2 )

edges of Bphas end vertices of degree

( p  1 )

and

p .

We observe that,

41 ) (

3

2

B

M

M

2

( B

4

)  142

M

2

( B

5

)  285

Proceeding like this, we obtain

) . 1 )(

2 2 ( . . 1 )]

1 )(

1 )[(

2 3 ( )

(

2

2

B p p p p p p p p p

M

p

        

for (p3).

3.3 Theorem: Let Fp be the friendship graph of order

p

where p2and the Second Zagreb index of the friendship graph is

. 8 4 )

(

2

2

F p p

M

p

 

Proof:

By the definition of Second Zagreb index, j

edges

i d

d M2

The friendship graph F2has 2 edges with end vertices of degree 2 and 2 and 4 edges with end vertices of degree 2 and 4.

The friendship graph

F

3has 3 edges with end vertices of degree 2 and 2 and 6 edges with end vertices of degree 2 and 6.

The friendship graph F4has 4 edges with end vertices of degree 2 and 2 and 8 edges with end vertices of degree 2 and 8.

For any

p

, the Friendship graph Fphas

p

edges with end vertices 2 and 2 and

2 p

edges with end vertices 2 and

2 p

. We observe that,

40 ) ( 2

2 F

M

M

2

( F

3

)  84

M

2

( F

4

)  144

Proceeding like this, the Second Zagreb Index of Fp for any p2is given by

. 8 4

) 2 . 2 )(

2 ( ) 2 . 2 ( ) (

2 2

p p

p p p

F M

p

IV. CONCLUSION

In this paper, the First and Second Zagreb indices for wheel graph, friendship graph and barbell graph were investigated.

This work would be extended to find the above indices for subdivision graphs.

References

1. Balaban A.T,Bonchev.d,Mekeyen.O,"Topological indices for structure-activity correlations,Topics Curr.Chem.Vol-114:21-55.

2. Chartrand G,Lesnial L,1986, “Graphs and digraphs”,Second edition, Wadsworth and Brooks/Cole, Montery, CA.

3. Gutman, Trinajstic.N, 1975,"Graph theory and molecular orbitals", Acyclic Polyenes, J.Chem.Phys.Vol-62:3399-3405.

(7)

Volume 2, Issue 10, October 2015 pg. 1-7

4. Gutman,Trinajstic,"Graphs and molecular orbitals",Chem.Phy.Lett.Vol-17:535-538.

5. https://en.wikipedia.org/wiki/Friendship_graph 6. https://en.wikipedia.org/wiki/Wheel_graph

7. Ivan Gutman,Kinkar Ch,2003,"The first Zagreb index after 30 years", MATCH Commun.Math.Comput.chem.Vol-50:83-92.

8. Kinkar Ch, Ivan Gutman, 2004,"Some properties of second Zagreb index", MATCH Commun.Math.Comput.chem.Vol -52:103-112.

9. Mathworld.wolfram.com/Barbell Graph.html

References

Related documents

The Chair shall preside at meetings; shall serve as the Workforce Board’s chief spokesperson and signatory; shall appoint committee chairs and committee members subject to these

linear regression model will all yield the same predicted value for that particular weld.

National Conference on Technical Vocational Education, Training and Skills Development: A Roadmap for Empowerment (Dec. 2008): Ministry of Human Resource Development, Department

• Follow up with your employer each reporting period to ensure your hours are reported on a regular basis?. • Discuss your progress with

Currently, National Instruments leads the 5G Test & Measurement market, being “responsible for making the hardware and software for testing and measuring … 5G, … carrier

4.1 The Select Committee is asked to consider the proposed development of the Customer Service Function, the recommended service delivery option and the investment required8. It

The corona radiata consists of one or more layers of follicular cells that surround the zona pellucida, the polar body, and the secondary oocyte.. The corona radiata is dispersed

We collected 15 species (60 samples) of Dendrobium from different locations in this investigation, which are the main commercial species in China.. The results of