The First and Second Zagreb Indices of Several Interesting Classes of Chemical Graphs and Nanostructures - Exotic Properties of Carbon Nanomatter - page 168

n n

deg H ( i )
u ∈ N G i ( w i )

+ deg G i ( u )deg G i ( v ) + deg G i ( u )

i = 1 uv ∈ E ( G i ) i = 1

n n

= M 2 ( H ) + M 2 ( G i ) + deg H ( i ) α G i ( w i )

i = 1 i = 1

+ + +

[ ω j deg H ( i ) ω i deg H ( j ) ω i ω j ] .

ij ∈ E ( H )

Suppose that w is the root vertex of a rooted graph K , and let G i = K and w i = w

∈{ }

for all i 1,2, ... , n . Using Theorem 7.5.1, we easily arrive at:

{ }

Corollary 7.5.2 The first and second Zagreb indices of the cluster H K are given

by:

1. M 1 ( H { K } ) = M 1 ( H ) + nM 1 ( K ) + 4 mω ,

+ m ( ω 2

2. M 2 ( H { K } ) = ωM 1 ( H ) + M 2 ( H ) + nM 2 ( K ) + 2 α K ( w )),

where ω

Let H be a labeled graph on n vertices. Choose a numbering for vertices of H
such that its pendant vertices have numbers 1,2, ... , k and its non-pendant vertices
have numbers k

= deg K ( w ) and m = | E ( H ) | .

+ 1, ... , n . Let G be a sequence of n rooted graphs G 1 , G 2 , ... , G n

with G i = K 1 for i ∈{ k + 1, k + 2 ... , n } . Using Theorem 7.5.1, we can easily get

the following result.

Corollary 7.5.3 The first and second Zagreb indices of the rooted product H ( G ) =

H ( G 1 , G 2 , ... , G k , K 1 , K 1 , ... , K 1 ) are given by:

i = 1 M 1 ( G i ) i = 1 ω i ,

k k

1. M 1 ( H ( G )) = M 1 ( H ) + + 2

i = 1 ω i α H ( i ),
and M 2 ( P 2 ( G 1 , G 2 )) = M 2 ( G 1 ) + M 2 ( G 2 ) + α G 1 ( w 1 ) + α G 2 ( w 2 ) + ( ω 1 + 1)( ω 2 + 1),
where for i

i = 1 M 2 ( G i ) + i = 1 α G i ( w i ) +

k k k

2. If H = P 2 , then M 2 ( H ( G )) = M 2 ( H ) +

∈{ 1,2, ... , k } , w i denotes the root vertex of G i and ω i denotes

its degree.

In the following corollary we consider a special case of Corollary 7.5.3, where
the components G i , i ∈{ 1,2, ... , k } , are mutually isomorphic to a rooted graph K.

Corollary 7.5.4 Let H be a labeled graph on n vertices whose pendant vertices have
numbers 1,2, ... , k and let K be a rooted graph with the root vertex w . Suppose G
is a sequence of n rooted graphs G 1 , G 2 , ... , G n with G i =

K for i ∈{ 1,2 ... , k } ,

and G i = K 1 for i ∈{ k + 1, k + 2 ... , n } . Then

1. M 1 ( H ( G )) = M 1 ( H ) + kM 1 ( K ) + 2 kω ,

ω i = 1 α H ( i ),

2. If H = P 2 , then M 2 ( H ( G )) = M 2 ( H ) + k ( M 2 ( K ) + α K ( w )) +

1) 2 , where ω

and M 2 ( P 2 ( G )) = 2 M 2 ( K ) + 2 α K ( w ) + ( ω + = deg K ( w ) .