ISSN: 2394-3122 (Online) Volume 2, Issue 10, October 2015
SK International Journal of Multidisciplinary Research Hub
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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 ( )
2vertices
d
iM
and jedges
i d
d
M2
withd
i stands for the degree (numberof 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 asW
p1orW
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 thanK4 W4, contain as a sub graph either
W
5 orW
6.
There is always a Hamiltonian cycle in the wheel graph and there are
( p
2 3 p 3 )
cycles inW
p[6].2.2 Theorem: Let
W
pbe the wheel graph of orderp
where p4 and the First Zagreb index of the wheel graph WpisVolume 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 p4 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 ofW
5 have degree 3 and 1 vertex ofW
5 have degree 4For
W
6, 5 vertices ofW
6 have degree 3 and 1 vertex ofW
6 have degree 5 We can observe that36 ) (
41
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 is2
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].Volume 2, Issue 10, October 2015 pg. 1-7
2.4 Theorem: Let
B
pbe the Barbell graph of orderp
, where (p3)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 p3 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 ofB
3 have degree 2 and 2 vertices ofB
3have degree 3 ForB4, 2*3 vertices of B4 have degree 3 and 2 vertices of B4 have degree 4 ForB
5, 2*4 vertices ofB
5 have degree 4 and 2 vertices ofB
5 have degree 5For a Barbell graphBp, 2(p1)vertices of Bphave degree (p1) 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 joiningp
copies of the cycle graph with a common vertex and it is denoted asF
p. F
p is a planar undirected graph with( 2 p 1 )
vertices and3 p
edges.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 orderp
where p2and 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 p2 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 ofF
3 have degree 2 and 1 center vertex ofF
3 have degree 6For
F
4, 8 vertices ofF
4 have degree 2 and 1 center vertex ofF
4 have degree 8 ForF
5, 10 vertices ofF
5 have degree 2 and 1 center vertex ofF
5 have degree 10For a Friendship graph
F
p, 2 p
vertices have degree 2 and 1 center vertex have degree2 p .
The First Zagreb Index of
F
2, M1(F2)4.2(22)32 We observe that,32 ) (
21
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 )
forp 2 .
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
wherep 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 p4 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)andv
i &vj are adjacentvertices.
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 )
forp 4 .
3.2 Theorem: Let Bpbe the Barbell graph of order
p
, where (p3)and the Second Zagreb index of the barbell graph is) . 1 )(
2 2 ( . . 1 )]
1 )(
1 )[(
2 3 ( )
(
22
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) ofB
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 endVolume 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) ofB
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 degreep
andp
and( 2 p 2 )
edges of Bphas end vertices of degree( p 1 )
and
p .
We observe that,
41 ) (
32
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 ( )
(
22
B p p p p p p p p p
M
p
for (p3).3.3 Theorem: Let Fp be the friendship graph of order
p
where p2and the Second Zagreb index of the friendship graph is. 8 4 )
(
22
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 Fphasp
edges with end vertices 2 and 2 and2 p
edges with end vertices 2 and2 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 p2is 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.
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