ZAGREB INDICES AND ZAGREB COINDICES OF SOME GRAPH OPERATIONS

http://www.iaeme.com/IJARET/index.asp 25 editor@iaeme.com
International Journal of Advanced Research in Engineering and Technology
(IJARET)
Volume 7, Issue 3, May–June 2016, pp. 25–41, Article ID: IJARET_07_03_003
Available online at
http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=7&IType=3
ISSN Print: 0976-6480 and ISSN Online: 0976-6499
© IAEME Publication
ZAGREB INDICES AND ZAGREB
COINDICES OF SOME GRAPH
OPERATIONS
K. Kiruthika
Assistant Professor, Mathematics,
Sri Shanmugha College of Engineering and Technology,
Salem [Dt] TamilNadu, India.
ABSTRACT
A topological index is the graph invariant numerical descriptor calculated
from a molecular graph representing a molecule. Classical Zagreb indices
and the recently introduced Zagreb coindices are topological indices related
to the atom – atom connectivity of the molecular structure represented by the
graph G. We explore here their basic mathematical properties and present
explicit formulae for these new graph invariants under several graph
operations.
Key words: Topological Index, Zagreb Indices and Zagreb Coindices, Graph
Operations.
Cite this article: K. Kiruthika. Zagreb Indices and Zagreb Coindices of Some
Graph Operations. International Journal of Advanced Research in
Engineering and Technology, 7(3), 2016, pp 25–41
http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=7&IType=3
1. INTRODUCTION
A representation of an object giving information only about the number of elements
composing it and their connectivity is named as topological representation of an
object. A topological representation of a molecule is called molecular graph. A
molecular graph is a collection of points representing the atoms in the molecule and
set of lines representing the covalent bonds. These points are named vertices and the
lines are named edges in graph theory language. The advantage of topological
indices is in that they may be used directly as simple numerical descriptors in a
comparison with physical, chemical or biological parameters of molecules in
Quantitative Structure Property Relationships (QSPR) and in Quantitative Structure
Activity Relationships (QSAR).
The first and second Zagreb indices [6] which are dependent on the degrees of
adjacent vertices appeared in the topological formula for the π-electron energy of

K. Kiruthika
conjugated systems, were first introduced by Gutman and Trinajstic (1983). The
Zagreb indices are found to have applications in QSPR and QSAR studies. Recently,
the first and second Zagreb coindices [1], a new pair of invariants, were introduced in
Doslic which are dependent on the degrees of non-adjacent vertices and thereby
quantifying a possible influence of remote pairs of vertices to the molecule’s
properties. Note that the Zagreb coindices of G are not agre indices of hile the
defining sums are over the set of edges of , the degrees are still with respect to G.
2. ZAGREB INDICES AND ZAGRED COINDICES
ll graphs considered are finite and simple et e a finite simple graph on n
vertices and m edges e denote the verte set and the edge set and
respectivel oslic he complement of denoted , is a simple graph on
the same set of vertices V(G) in which two vertices u and v are adjacent. (i.e)
connected by an edge uv, iff they are not adjacent in G. Hence
( ) ( )uv E G uv E G   . Obviously, ( ) ( ) ( )nE G E G E K and
( )
2
n
m E G m
 
   
 
.The degree of a vertex u in G is denoted by ( )Gd u the degree
of the same verte in is then given ( ) 1 ( )GG
d u n d u   .The sum of the degrees
of vertices of a graph G is twice the number of edges. (i.e)
( )
( ) 2G
u V G
d u m

 .
2.1. ZAGREB INDICES
The First Zagreb Index of G [5] is denoted by 1( )M G and is defined as
 2
1
( )
( ) ( )G
u V G
M G d u

 
.
The first Zagreb index can also be expressed as a sum over edges in G.
 1
( )
( ) ( ) ( )G G
uv E G
M G d u d v

 
The Second Zagreb Index of G is denoted by 2( )M G and is defined as
2
( )
( ) ( ) ( )G G
uv E G
M G d u d v

 

Zagreb Indices and Zagreb Coindices of Some Graph Operations
Example1
Chemical compound: Methylcyclopropane
Figure 1
Here 1 2 3 4, , ,v v v v represents the labeling of the graph G, while the underlined
numbers stand for the valencies of the vertices in G.
2 2 2 2
1( ) 1 3 2 2 18M G     
2( ) 1.3 3.2 3.2 2.2 19M G     
2.2 ZAGREB COINDICES
The First Zagreb Coindex[1]of G is denoted by 1( )M G and is defined as
 1
( )
( ) ( ) ( )G G
uv E G
M G d u d v

 
The Second Zagreb Coindex of G is denoted by 2( )M G and is defined as

K. Kiruthika
2
( )
( ) ( ) ( )G G
uv E G
M G d u d v

 
EXAMPLE 2
Chemical compound: Methylcyclopentane
Figure 2
Here ( )E G = {v1v5, v1v6, v1v3, v1v4, v2v5, v2v6, v3v4, v3v6, v4v5}
1( )M G = 34 and 2( )M G = 32
3 BASIC PROPERTIES OF ZAGREB COINDICES
Proposition 1
Let G be a simple graph [4] on n vertices and m edges. Then
1 1( ) ( ) 2( 1)( )M G M G n m m    .

Proof
We know that
2
1
( )
( ) ( )G
u V G
M G d u

 
2
1
( )
( ) ( )G
u V G
M G d u

  =
 
2
( )
1 ( )G
u V G
n d u

 
=
 
2 2
( ) ( ) ( )
1 2( 1) ( ) ( )G G
u V G u V G u V G
n n d u d u
  
     
2
1( 1) 4 ( 1) ( )n n m n M G    
1( )M G = 12( 1)( ) ( )n m m M G  
Proposition2
Let G be a simple graph on n vertices and m edges. Then 1 1( ) 2 ( 1) ( )M G m n M G   .
Proof
The First Zagreb Coindex of G is
 1
( )
( ) ( ) ( )G G
uv E G
M G d u d v

 
( )
1 ( ) 1 ( )G G
uv E G
n d u n d v

        
( )
2( 1) ( ( ) ( ))G G
uv E G
n d u d v

      
12 ( 1) ( )m n M G  
 
2
( )
2
( )
2( 1) ( )
2
2( 1) ( 1 ( )
2
G
u V G
G
u V G
n
n m d u
n
n m n d u


  
     
  
  
       
  


2 2
( ) ( ) ( )
( 1)
2( 1) ( 1) 2( 1) ( ) ( )
2
G G
u V G u V G u V G
n n
n m n n d u d u
  
  
            
  
2 2
1( 1) 2 ( 1) ( 1) 4 ( 1) ( )n n m n n n m n M G        
1 1( ) 2 ( 1) ( )M G m n M G  
Proposition 3
Let G be a simple graph on n vertices and m edges. Then
2
2 2 1( ) ( ) ( 1) ( ) ( 1)M G M G n M G m n    

K. Kiruthika
Proof
The Second Zagreb Coindex is 2
( )
( ) ( ) ( )G G
uv E G
M G d u d v

 
  
( )
( 1) ( ) ( 1) ( )G G
uv E G
n d u n d v

    
2
( ) ( ) ( )
( 1) ( 1) ( ) ( ) ( ) ( )G G G G
uvE G uvE G uvE G
n n d u d v d u d v       
2( )M G
2
2 1( ) ( 1) ( ) ( 1)M G n M G m n    
Proposition 4
Let G be a simple graph on n vertices and m edges. Then 11( ) ( )M G M G .
Proof
By applying proposition 2 to G , we get
1 1( ) 2 ( 1) ( )M G m n M G  
Now, plugging in the expression for 1( )M G from proposition 1 yields
1 1( ) 2 ( 1) ( ) 2( 1)( )M G m n M G n m m     
12 ( 1) ( )m n M G  
11( ) ( )M G M G .
Proposition 5
Let G be a simple graph on n vertices and m edges. Then
2
2 2 1
1
( ) 2 ( ) ( )
2
M G m M G M G   .
Proof
The result follows by squaring both sides of the identity
( )
( ) 2G
u V G
d u m

 and then
Splitting the term 2 ( ) ( )G Gd u d v into two sums, one over the edges of G and the
other over the edges of G (Gutman 2004).
2 2
( ) ( ) ( )
2 2
( ) ( )( )
2 2
( ) ( )( )
( ) 2 ( ) ( ) 2 ( ) ( ) 4
2 ( ) ( ) 4 ( ) 2 ( ) ( )
1
( ) ( ) 2 ( ) ( ) ( )
2
G G G G G
u V G uv E G uv E G
G G G G G
u V G uv E Guv E G
G G G G G
u V G uv E Guv E G
d u d u d u d u d u m
d u d u m d u d u d u
d u d u m d u d u d u
  
 
 
  
  
  
  
  
  

2
2 1 2
1
( ) 2 ( ) ( )
2
M G m M G M G  
4. MAIN RESULTS
In this section weintroduce the Zagreb indices and coindicesof Cartesian product,
Disjunction, Composition, Tensor product and Normal product of graphs. All
considered operations are binary. Hence, we usually deal with two finite and simple
graphs, 1G and 2G . For a given graph iG , its vertex set and edge set will be denoted by
iV and iE respectively, where i = 1, 2. The number of edges in iG is denoted by im
.When more than two graphs can be combined using a given operation, the values of
subscripts will vary accordingly.
4.1 CARTESIAN PRODUCT
For given graphs 1G and 2G , (Ashrafi 2009) Cartesian product of 1 2G G is the graph
on the vertex sets 1 2V V and 1 1( , )u v and 2 2( , )u v are adjacent iff 1 2u u and 1v is
adjacent to 2v or 1u is adjacent to 2u and 1 2v v . The degree of a vertex 1 1( , )u v of
1 2G G is given by 1 2 1 21 1 1 1( , ) ( ) ( )  G G G Gd u v d u d v .
4.2 DISJUNCTION
The disjunction 1 2G G of two graphs 1G and 2G is the graph [2] with the vertex sets
1 2V V and 1 1( , )u v and 2 2( , )u v are adjacent iff 1u is adjacent to 2u or 1v is adjacent to
2v . The degree of a vertex 1 1( , )u v of 1 2G G is given by
1 2 1 2 1 21 1 2 1 1 1 1 1( , ) ( ) ( ) ( ) ( )   G G G G G Gd u v n d u nd v d u d v
.
4.3 COMPOSITION
The composition  1 2G G of two graphs 1G and 2G is the graph with the vertex sets
1 2V V and 1 1( , )u v and 2 2( , )u v are adjacent iff [ 1u is adjacent to 2u ] or [ 1 2u u and
1v is adjacent to 2v ]. The degree of a vertex 1 1( , )u v of
 1 2G G is given by
  1 21 2 1 1 2 1 1( , ) ( ) ( )G GG G
d u v n d u d v  .
4.4 TENSOR PRODUCT
Let 1 1 1( , )G V E and 2 2 2( , )G V E be twographs whose vertex sets are disjoint as also
their edge sets (Parthasarathy 1994) [7]. The tensor product 1 2G G G  of graphs 1G
and 2G has as its vertex set 1 2V V and 1 1( , )u v and 2 2( , )u v are adjacent iff 1u is

K. Kiruthika
adjacent to 2u and 1v is adjacent to 2v . The degree of a vertex ( , )i ju v of 1 2G G is
given by 1 2 1 2
( , ) ( ) ( )G G i j G i G jd u v d u d v  .
4.5 NORMAL PRODUCT
The normal product 1 2G G G (strong product or wreath product) of two graphs
1G and 2G (BaskarBabujee 2012) has as its vertex set 1 2V V and 1 1( , )u v and
2 2( , )u v are adjacent iff 1u is adjacent to 2u and 1 2v v or
1 2u u and 1v is adjacent to 2v or
1u is adjacent to 2u and 1v is adjacent to 2v .The degree of a vertex ( , )i ju v of
1 2G G is given by  1 2 1 2 1 2
( , ) ( ) ( ) ( ) ( )G G i j G i G j G i G jd u v d u d v d u d v   .
All graphs are finite and simple. We denote the vertex and edge sets of G by V(G)
and E(G) respectively. The number of vertices and edges of G by ( )V G n and
( )E G m respectively. It can be easily seen that
 1 2 1 2 1 2( ) ( ) ( )V G G V G G V G G   
1 2 1 2( ) ( )V G G V G G   1 2( ) ( )V G V G
LEMMA 6 If 1G and 2G are two simple graphs then,
1 2 2 1 1 2( ) ( ) ( ) ( ) ( )  E G G V G E G V G E G
2 2
1 2 2 1 1 2 1 2( ) ( ) ( ) ( ) ( ) 2 ( ) ( )   E G G V G E G V G E G E G E G
  2
1 2 2 1 1 2( ) ( ) ( ) ( ) ( )E G G V G E G V G E G 
1 2 1 2( ) 2 ( ) ( )E G G E G E G 
1 2 1 2 2 1 1 2( ) 2 ( ) ( ) ( ) ( ) ( ) ( )E G G E G E G V G E G V G E G  
PROOF
The part (a-e) are consequence of definitions and some wellknown results of the book
of Imrich and Klavzar [3].
ZAGREB TYPE INDICES FOR CARTESIAN PRODUCT OF GRAPHS
THEOREM 7
The First Zagreb index of Cartesian product of two graphs 1G and 2G is
1 1 2 2 1 1 1 1 2 1 2( ) ( ) ( ) 8M G G n M G n M G mm   

PROOF
1 1 2( )M G G
1 2
1 2
2
1 1
( , )
 
   
n n
G G i j
i j
d u v
Using the definition of Cartesian product, we get
1 2
1 2
2
1 1
( ) ( )
 
   
n n
G i G j
i j
d u d v
1 2
1 2 1 2
2 2
1 1
( ) ( ) 2 ( ) ( )
 
    
n n
G i G j G i G j
i j
d u d v d u d v
1 2 1 2 1 2
1 2 1 2
2 2
1 1 1 1 1 1
( ) ( ) 2 ( ) ( )
n n n n n n
G i G j G i G j
i j i j i j
d u d v d u d v
     
     
2 1 1 1 1 2 1 2( ) ( ) 8  n M G n M G mm
THEOREM 8
The First Zagreb coindex of cartesian product of two graphs 1G and 2G is
 1 1 2 1 2 1 2 2 1 1 2 2 1 1 1 2 2( ) 2 ( 1) ( 1) 8 ( ) ( )M G G n n m n m n mm n M G n M G        .
PROOF: By Lemma (6-a) and Theorem 7,
Using the property 1 1( ) 2 ( 1) ( )M G m n M G  
1 1( ) 2 ( ) ( ( ) 1) ( )M G E G V G M G  
 1 1 2 1 2 1 2 1 1 2( ) 2 ( ) ( ) 1 ( )      M G G E G G V G G M G G
 2 1 1 2 1 2 1 1 22 ( 1) ( )    n m n m n n M G G
1 2 2 1 1 2 1 2 2 1 1 1 1 22( )( 1) 8 ( ) ( )n m n m n n mm n M G n M G     
1 2 2 1 1 2 1 2 2 1 1 1 1 1 2 2 1 22( 1)( ) 8 2 ( 1) ( ) 2 ( 1) ( )n n n m n m m m n m n M G n m n M G               
1 2 2 1 1 2 2 1 1 2 1 2 1 1 2
1 2 2 1 1 1 2 2 2 1 1 1 2
2 ( ) 2( ) 8 2
2 ( ) 2 2 ( )
n n n m n m n m n m m m m n n
m n n M G n n m m n n M G
     
    
 1 1 2 1 2 1 2 2 1 1 2 2 1 1 1 2 2( ) 2 ( 1) ( 1) 8 ( ) ( )M G G n n m n m n mm n M G n M G        .

K. Kiruthika
ZAGREB TYPE INDICES FOR DISJUNCTION OF GRAPHS
THEOREM 9
The First Zagreb index of disjunction of two graphs 1G and 2G is
2 2
1 1 2 2 2 2 1 1 1 1 1 1 2 1 2 1 2 1 1 1 2( ) ( 4 ) ( ) ( 4 ) ( ) 8 ( ) ( )      M G G n m n M G n m n M G mm nn M G M G
PROOF:
Using the definition of disjunction, we get
1 1 2( )M G G
1 2
1 2
2
1 1
( , )
n n
G G i j
i j
d u v
 
   
 
1 2
1 2 1 2
2
2 1
1 1
( ) ( ) ( ) ( )
n n
G i G j G i G j
i j
n d u n d v d u d v
 
   

1 2
1 2 1 2
2 2
2 1
1 1
( ) ( ) ( ) ( )
n n
G i G j G i G j
i j
n d u n d v d u d v
 
        
1 2 1 22 12 ( ) ( ) ( ) ( )G i G j G i G jn d u n d v d u d v       
1 2 1 2 1 2
1 2 1 2
2 2 2 2
2 1 1 2
1 1 1 1 1 1
( ) ( ) 2 ( ) ( )
n n n n n n
G i G j G i G j
i j i j i j
n d u n d v n n d u d v
     
     
1 2 1 2
1 2 1 2
2 2 2
2
1 1 1 1
( ) ( ) 2 ( ) ( )
n n n n
G i G j G i G j
i j i j
d u d v n d u d v
   
    
1 2
1 2
2
1
1 1
2 ( ) ( )
n n
G i G j
i j
n d u d v
 
  
3 3
2 1 1 1 1 2 1 2 1 2 1 1 1 2( ) ( ) 2 (2 )(2 ) ( ) ( )n M G n M G nn m m M G M G   
2 1 1 2 1 1 1 22 ( )(2 ) 2 (2 ) ( )n M G m n m M G 
2 2
1 1 2 2 2 2 1 1 1 1 1 1 2 1 2 1 2 1 1 1 2( ) ( 4 ) ( ) ( 4 ) ( ) 8 ( ) ( )      M G G n m n M G n m n M G mm nn M G M G
THEOREM 10
The First Zagreb coindex of disjunction of two graphs 1G and 2G is
2 2
1 1 2 1 2 2 2 1 1 1 2 1 2( ) 2 ( 1) 2 ( 1) 4 ( )M G G mn n m n n mm n n      
3 3
2 2 2 2 1 1 1 1 1 1 1 22 2 ( ) 2 2 ( )n m n m M G n mn m M G             1 1 1 2( ) ( )M G M G
PROOF: By Lemma (6-b) and Theorem 9,
1 1( ) 2 ( ) ( ( ) 1) ( )M G E G V G M G  

 1 1 2 1 2 1 2 1 1 2( ) 2 ( ) ( ) 1 ( )M G G E G G V G G M G G      
 2 2
2 1 1 2 1 2 1 2 1 1 22 2 ( 1) ( )n m n m mm n n M G G     
3 3 2 2
1 2 1 1 2 2 1 2 1 2 1 2 2 1 1 22 2 4 2 2 4nn m n n m mm nn mn m n mm     
2 2
2 2 2 1 1 1 1 1 1 2 1 2 1 2 1 1 1 2( 4 ) ( ) ( 4 ) ( ) 8 ( ) ( )n m n M G n m n M G mm n n M G M G     
3 3 2 2
1 2 1 2 1 2 1 2 1 1 2 2 2 1 1 24 12 2 2 2 2mm mm nn nn m n n m n m n m     
3 3
2 2 2 1 1 1 1 1 1 1 2 2 1 2(4 ) 2 ( 1) ( ) (4 ) 2 ( 1) ( )m n n m n M G m n n m n M G             
1 1 1 1 2 2 1 22 ( 1) ( ) 2 ( 1) ( )m n M G m n M G          
2 2
1 1 2 1 2 2 2 1 1 1 2 1 2( ) 2 ( 1) 2 ( 1) 4 ( )M G G mn n m n n mm n n      
3 3
2 2 2 2 1 1 1 1 1 1 1 22 2 ( ) 2 2 ( )n m n m M G n mn m M G             1 1 1 2( ) ( )M G M G
ZAGREB TYPE INDICES FOR COMPOSITION OF GRAPHS
THEOREM 11
The First Zagreb index of composition of two graphs 1G and 2G is
  3
1 1 2 2 1 1 1 1 2 2 1 2( ) ( ) ( ) 8M G G n M G n M G n mm  
PROOF:
Using the definition of Composition, we get
 1 1 2( )M G G
 
1 2
1 2
2
1 1
( , )
n n
i jG G
i j
d u v
 
   
1 2
1 2
2
2
1 1
( ) ( )
n n
G i G j
i j
n d u d v
 
   
1 2
1 2 1 2
2 2
2 2
1 1
( ) ( ) 2 ( ) ( )
n n
G i G j G i G j
i j
n d u d v n d u d v
 
    
1 2 1 2 1 2
1 2 1 2
2 2
2 2
1 1 1 1 1 1
( ) ( ) 2 ( ) ( )
n n n n n n
G i G j G i G j
i j i j i j
n d u d v n d u d v
     
     
  3
1 1 2 2 1 1 1 1 2 2 1 2( ) ( ) ( ) 8M G G n M G n M G n mm  
THEOREM 12
The First Zagreb coindex of composition of two graphs 1G and 2G is
  2 3
1 1 2 1 2 2 1 2 1 2 2 1 2 2 1 1 1 1 2( ) 2 ( 1) 2 ( 1) 8 ( ) ( )M G G n n m n n m n n mm n M G n M G      
PROOF: By Lemma (6 – c) and Theorem 11,

K. Kiruthika
1 1( ) 2 ( ) ( ( ) 1) ( )M G E G V G M G  
        1 1 2 1 2 1 2 1 1 2( ) 2 ( ) ( ) 1 ( )M G G E G G V G G M G G  
 2
2 1 1 2 1 2 1 1 22 ( 1) ( )n m n m n n M G G    
2 2
1 2 1 2 2 1 1 2 2 12 ( ) 2( )nn nm n m nm n m    3
2 1 2 2 1 1 1 1 28 ( ) ( )n mm n M G n M G  
2 2
1 2 1 2 2 1 1 2 1 2 2 1 22 ( ) 2( ) 8nn nm n m mn nm n mm    
3
2 1 1 1 1 1 2 2 1 22 ( 1) ( ) 2 ( 1) ( )n m n M G n m n M G           
From the above, after simple calculations, the desired results follows
  2 3
1 1 2 1 2 2 1 2 1 2 2 1 2 2 1 1 1 1 2( ) 2 ( 1) 2 ( 1) 8 ( ) ( )M G G n n m n n m n n mm n M G n M G      
ZAGREB TYPE INDICES FOR TENSOR PRODUCT OF GRAPHS
THEOREM 13
If 1G and 2G are two simple graphs and if 1 2G G is connected then first Zagreb
index of 1 2G G is 1 1 2 1 1 1 2( ) ( ) ( )M G G M G M G  .
PROOF:
The first Zagreb index of 1 2G G is
1 1 2( )M G G
1 2
1 2
2
1 1
( , )
n n
G G i j
i j
d u v
 
   
1 2
1 2
1
1 1 2
1
1
2
1 1
2 2 2
1
1
2
1 2
1
( ) ( )
( ( )) ( ( )) ....... (( ( ))
( ( )) ( )
n n
G i G j
i j
n
G i G G j
i
n
G i
i
d u d v
d u d v d v
d u M G
 


   
    




1 1 2 1 1 1 2( ) ( ) ( )M G G M G M G 
THEOREM 14
If 1G and 2G are two simple graphs and if 1 2G G is connected then second Zagreb
index of 1 2G G is 2 1 2 2 1 2 2( ) 2 ( ) ( )M G G M G M G  .

PROOF
The second Zagreb index of 1 2G G is
1 2 1 2
1 1 2 2 1 2
2 1 2 ( ) 1 1 ( ) 2 2
( , )( , ) ( )
( ) ( , ) ( , )G G G G
u v u v E G G
M G G d u v d u v 
 
  
1 2 1 2
1 1 2 2 1 2
1 1 2 2
1 2 1 1 2 1
1 1 2 2
( , )( , ) ( )
1 2 1 2
( , ) ( ) ( , ) ( )
( ) ( ) ( ) ( )
2 ( ) ( ) ( ) ( )
G G G G
u v u v E G G
G G G G
u u E G u u E G
d u d v d u d v
d u d u d v d v
 
 



 
2 1 2 2 1 2 2( ) 2 ( ) ( )M G G M G M G 
THEOREM 15
If 1G and 2G are two simple graphs and if 1 2G G is connected then first Zagreb
coindex of 1 2G G is 1 1 2 1 2 1 2 2 2 1 1( ) 4 ( 2) 2 ( 1) ( )M G G mm n n m n M G     
1 1 1 2 1 1 1 22 ( 1) ( ) ( ) ( )m n M G M G M G  
PROOF: By Lemma (6 – d) and Theorem 13,
We know that 1 1( ) 2 ( 1) ( )M G m n M G  
1 1( ) 2 ( ) ( ( ) 1) ( )M G E G V G M G  
 1 1 2 1 2 1 2 1 1 2( ) 2 ( ) ( ) 1 ( )M G G E G G V G G M G G      
 1 2 1 1 1 1 1 22 2 ( ) ( ) ( ) ( ) 1 ( ) ( )E G E G V G V G M G M G    
 1 2 1 2 1 1 1 24 1 ( ) ( )mm n n M G M G  
 1 2 1 24 1mm n n  1 1 1 1 2 2 1 22 ( 1) ( ) 2 ( 1) ( )m n M G m n M G          
1 2 1 2 1 2 1 2
1 1 1 2 2 2 1 1 1 1 1 2
1 2 1 2 1 2 1 2
1 1 1 2 2 2 1 1 1 1 1 2
4 ( 1) 4 ( 1)( 1)
2 ( 1) ( ) 2 ( 1) ( ) ( ) ( )
4 ( 1 1)
2 ( 1) ( ) 2 ( 1) ( ) ( ) ( )
m m n n m m n n
m n M G m n M G M G M G
m m n n n n n n
m n M G m n M G M G M G
    
     
     
     
1 1 2 1 2 1 2 2 2 1 1( ) 4 ( 2) 2 ( 1) ( )M G G mm n n m n M G     
1 1 1 2 1 1 1 22 ( 1) ( ) ( ) ( )m n M G M G M G  

K. Kiruthika
THEOREM 16
Let 1G and 2G be two simple graphs then, 1 2G G is connected such that second
Zagrebcoindex of 1 2G G is
2 1 2 2 2 1 2 2 1 2 2 1 1( ) 2 ( )(2 1) 2 ( )(2 1)M G G m M G m n m M G m n      
2 1 1 2 2 1 1 2 1 1 2 1 2 22 ( )( 1) 2 ( )( 1) 2 ( ) ( )m M G n m m M G n m M G M G      
1 2 1 2 2 1 1 2 1 2 2 1
1 1 2 2 1 1 1 2
4 ( ( 1) ( 1) ( 1)( 1)) ( ) ( )
( ) ( ) ( ) ( )
m m m n m n n n M G M G
M G M G M G M G
       
 
PROOF: By Lemma (6 – d), Theorem 13 and Theorem 14,
We know that
2
2 2 1
1
( ) 2 ( ) ( )
2
M G m M G M G  
.
2
2 2 1
1
( ) 2 ( ) ( ) ( )
2
M G E G M G M G  
.
2
2 1 2 1 2 2 1 2 1 1 2
1
( ) 2 ( ) ( ) ( )
2
M G G E G G M G G M G G      
2
1 2 2 1 2 2 1 1 1 2
1
2 2 ( ) ( ) 2 ( ) ( ) ( ) ( )
2
E G E G M G M G M G M G    
2
1 2 2 1 2 2 1 1 1 2
1
2(2 ) 2 ( ) ( ) ( ) ( )
2
m m M G M G M G M G  
2 2 2
1 2 1 2 1 1 1
1
8 2 2 ( ) ( )
2
m m m M G M G
 
     
2
2 2 2 1 2
1
2 ( ) ( )
2
m M G M G
 
   
1 1 1 1 2 2 1 2
1
2 ( 1) ( ) 2 ( 1) ( )
2
m n M G m n M G        
 2 2 2
1 2 1 2 1 1 1 1 1
1
8 2 2 ( ) 2 ( 1) ( )
2
m m m M G m n M G
 
       
 2
2 2 2 2 2 1 2
1
2 ( ) 2 ( 1) ( )
2
m M G m n M G
 
     
1 1 1 1 2 2 1 2
1
2 ( 1) ( ) 2 ( 1) ( )
2
m n M G m n M G        
By above calculations, we conclude that
2 1 2 2 2 1 2 2 1 2 2 1 1( ) 2 ( )(2 1) 2 ( )(2 1)M G G m M G m n m M G m n      
2 1 1 2 2 1 1 2 1 1 2 1 2 22 ( )( 1) 2 ( )( 1) 2 ( ) ( )m M G n m m M G n m M G M G      

1 2 1 2 2 1 1 2 1 2 2 1
1 1 2 2 1 1 1 2
4 ( ( 1) ( 1) ( 1)( 1)) ( ) ( )
( ) ( ) ( ) ( )
m m m n m n n n M G M G
M G M G M G M G
       
 
ZAGREB TYPE INDICES FOR NORMAL PRODUCT OF GRAPHS
THEOREM 17
The First Zagreb index of normal product of two graphs 1G and 2G is
1 1 2 1 1 1 2 2 2 1 1 1 2 1 1 1 2( ) (4 ) ( ) (4 ) ( ) 8 ( ) ( )M G G m n M G m n M G mm M G M G     
PROOF
 
1 2
1 2 1 2
2
1 1 2
1 1
( ) ( ) ( ) ( ) ( )
n n
G i G j G i G j
i j
M G G d u d v d u d v
 
   

1 2
1 2 1 2
2 2
1 1
( ) ( ) ( ) ( )
n n
G i G j G i G j
i j
d u d v d u d v
 
        
1 2 1 2
2 ( ) ( ) ( ) ( )G i G j G i G jd u d v d u d v       

1 2
1 2 1 2
2 2
1 1
( ) ( ) 2 ( ) ( )
n n
G i G j G i G j
i j
d u d v d u d v
 
    
1 2 1 2 1 2
2 2 2 2
( ) ( ) 2 ( ) ( ) 2 ( ) ( )G i G j G i G j G i G jd u d v d u d v d u d v  
1 2 1 2 1 2
1 2 1 2
2 2
1 1 1 1 1 1
( ) ( ) 2 ( ) ( )
n n n n n n
G i G j G i G j
i j i j i j
d u d v d u d v
     
     
1 2 1 2
1 2 1 2
2 2 2
1 1 1 1
( ) ( ) 2 ( ) ( )
n n n n
G i G j G i G j
i j i j
d u d v d u d v
   
    
1 2
1 2
2
1 1
2 ( ) ( )
n n
G i G j
i j
d u d v
 
  
2 1 1 1 1 2 1 2 1 1 1 2( ) ( ) 2(2 )(2 ) ( ) ( )n M G n M G m m M G M G   
1 1 2 1 1 22 ( )(2 ) 2(2 ) ( )M G m m M G 
2 1 1 1 1 2 1 2 1 1 1 2( ) ( ) 8 ( ) ( )n M G n M G mm M G M G    2 1 1 1 1 24 ( ) 4 ( )m M G m M G 
1 1 2 1 1 1 2 2 2 1 1 1 2 1 1 1 2( ) (4 ) ( ) (4 ) ( ) 8 ( ) ( )M G G m n M G m n M G mm M G M G     
THEOREM 18
The First Zagreb coindex of normal product of two graphs 1G and 2G is
1 1 2 1 2 1 2 1 2 2 1 1 2 1 2 2 2 1 1( ) 2 ( 1) 2 ( 1) 4 ( ) (4 ) ( )M G G n n m n n n m n mm n n m n M G       
1 1 1 2 1 1 2 2 2 2 1 2 1 1 1 1( ) ( ) ( )( 2 2 ) ( )( 2 2 )M G M G M G n m n m M G n mn m      

K. Kiruthika
PROOF: By Lemma (6 – e) and Theorem 17,
1 1( ) 2 ( ) ( ( ) 1) ( )M G E G V G M G  
 1 1 2 1 2 1 2 1 1 2( ) 2 ( ) ( ) 1 ( )M G G E G G V G G M G G  
 1 2 1 2 2 1 1 2 1 1 22 2 ( 1) ( )mm n m n m n n M G G    
1 1 2( )M G G 2 2
1 2 1 2 1 2 1 2 2 1 2 1 2 14 4 2 2 2mm nn mm n n m nm nn m     2 12n m
 1 1 1 2 2 2 1 1(4 ) ( ) (4 ) ( )m n M G m n M G    1 2 1 1 1 28 ( ) ( )mm M G M G 
2 2
1 2 1 2 1 2 1 2 2 1 2 1 2 14 4 2 2 2mm nn mm n n m nm nn m     2 12n m
1 1 2 2 1 2(4 ) 2 ( 1) ( )m n m n M G      2 2 1 1 1 1(4 ) 2 ( 1) ( )m n m n M G     
1 28m m 1 1 1 1 2 2 1 22 ( 1) ( ) 2 ( 1) ( )m n M G m n M G          
2 2
1 2 1 2 1 2 1 2 2 1 2 1 2 14 4 2 2 2mm nn mm n n m nm nn m     2 1 1 2 22 8 ( 1)n m mm n  
1 1 2 1 2 24 ( ) 2 ( 1)m M G n m n   1 1 2 1 2 1 2 1 1( ) 8 ( 1) 4 ( )n M G mm n m M G   
2 1 12 ( 1)n m n  2 1 1 1 2 1 2 1 2( ) 8 4 ( 1)( 1)n M G mm mm n n    
2 2 1 1 1 1 1 22 ( 1) ( ) ( ) ( )m n M G M G M G  
1 1 2 1 2 1 2 1 2 2 1 1 2 1 2 2 2 1 1( ) 2 ( 1) 2 ( 1) 4 ( ) (4 ) ( )M G G n n m n n n m n mm n n m n M G       
1 1 1 2 1 1 2 2 2 2 1 2 1 1 1 1( ) ( ) ( )( 2 2 ) ( )( 2 2 )M G M G M G n m n m M G n mn m      
CONCLUSION
In this paper, we have studied the Zagreb indices and Zagreb coindices andalso we
have investigated their basic mathematical properties and obtained explicit formulae
for computing their values under several graph operations namely Cartesian product,
Disjunction, Composition, Tensor product and Normal product of graphs.
Nevertheless, there are still many other graph operations and special classes of graphs
that are not covered here and we mention some possible directions for future research.

REFERENCES
[1] Ashrafi A R, T Doslic, A Hamzeb, The Zagreb coindices of Graph
operations, Discrete Applied Mathematics 158 (2010) 1571-1578.
[2] BaskarBabujee J, Ramakrishnan S, Zagreb indices and coindices for
compound graphs, proceedings of ICMCM 2012, PSG Tech 19-21 Dec 2011,
Narosa publications, pp: 357–362.
[3] W. Imrich, S. Klavzar, Product Graphs: Strudture and Recognition, John
Wiley & Sons, New York, USA 2000.
[4] Das K Ch, I Gutman, Some properties of the second Zagreb Index, MATCH
Commun. Math. Comput. Chem 52 (2004) 103-112.
[5] Khalifeh M H, H Yousefi-Azari, A R Ashrafi, The First and Second Zagreb
indices of some graph operations, Discrete Applied Mathematics 157 (2009)
804–811
[6] Nenad Trinajstic, Chemical Graph Theory, Volume II, CRC Press, Inc., Boca
Raton, Florida, 1983.
[7] Parthasarathy K R, Basic Graph Theory, Tata McGraw Hill Publishing
Company Limited, New Delhi, 1994.

ZAGREB INDICES AND ZAGREB COINDICES OF SOME GRAPH OPERATIONS

Recommended

Recommended

More Related Content

Similar to ZAGREB INDICES AND ZAGREB COINDICES OF SOME GRAPH OPERATIONS

Similar to ZAGREB INDICES AND ZAGREB COINDICES OF SOME GRAPH OPERATIONS (20)

More from IAEME Publication

More from IAEME Publication (20)

Recently uploaded

Recently uploaded (20)

ZAGREB INDICES AND ZAGREB COINDICES OF SOME GRAPH OPERATIONS