The First and Second Zagreb Indices of Several Interesting Classes of Chemical Graphs and Nanostructures - Exotic Properties of Carbon Nanomatter - page 159

Suppose that v and w are two vertices of a graph G , and let G i = G , v i = v , and

w i = w for all i ∈{ 1,2, ... , d } . Then Theorem 7.3.2 implies,

Corollary 7.3.3 The first Zagreb index of the bridge graph B 2 ,( d ≥ 2 times), is

given by:

M 1 ( B 2 ) = dM 1 ( G ) + 2( d − 1)( υ + ω + 1),

where υ = deg G ( v ), ω = deg G ( w ) .

Theorem 7.3.4 The second Zagreb index of the bridge graph B 2 , d ≥ 2. is given by:

d d − 1 d

M 2 ( B 2 ) = M 2 ( G i ) + [ ω i + α G i ( w i )] + [ υ i + α G i ( v i )]

i = 1 i = 1 i = 2

d − 1

+ ω i υ i + 1 + d + n − 1,

i = 1

where υ i = deg G i ( v i ), ω i = deg G i ( w i ), for 1 ≤ i ≤ d and n is the number of the

graphs G i ,2 ≤ i ≤ d − 1, with the property that the vertices v i and w i are adjacent

in them.

Proof From the definition of the bridge graph B 2 , E ( B 2 ) =

E ( G 1 ) E ( G 2 ) ... E ( G d ) {

. Using the same
argument as in the proof of Theorem 7.2.4, we partition the sum in the formula of
M 2 ( B 2 ) into the five sums as follows:
The first sum S 1 is taken over all edges ab

w i v i + 1 | 1 ≤ i ≤ d − 1 }

∈ E ( G 1 ). Using Lemma 7.3.1, we

have:

S 1 = deg B 2 ( a )deg B 2 ( b ) = deg G 1 ( a )deg G 1 ( b )

ab ∈ E ( G 1 ) ab ∈ E ( G 1 ); a , b = w 1

+ deg G 1 ( a )[deg G 1 ( w 1 ) + 1]

ab ∈ E ( G 1 ); a ∈ V ( G 1 ), b = w 1

= M 2 ( G 1 ) + α G 1 ( w 1 ) .

Analogously,

S 2 = deg B 2 ( a )deg B 2 ( b ) = deg G d ( a )deg G d ( b )

ab ∈ E ( G d ) ab ∈ E ( G d ); a , b = v d

+ deg G d ( a )[deg G d ( v d ) + 1]

ab ∈ E ( G d ); a ∈ V ( G d ), b = v d

= M 2 ( G d ) + α G d ( v d ) .

Suppose I . The third sum S 3 is taken over all

edges ab ∈ E ( G i ) for all i ∈ I . Using Lemma 7.3.1, we have:

={ i | 2 ≤ i ≤ d − 1, v i w i ∈ E ( G i ) }

S 3 = deg B 2 ( a )deg B 2 ( b )

i ∈ I ab ∈ E ( G i )