The First and Second Zagreb Indices of Several Interesting Classes of Chemical Graphs and Nanostructures - Exotic Properties of Carbon Nanomatter - page 158

Fig. 7.5 The bridge graph B 2 = B 2 ( G 1 , G 2 , ... , G d ; v 1 , w 1 , v 2 , w 2 , ... , v d , w d )

First, we express the following simple lemma. It follows immediately from the
definition of B 2 .
Lemma 7.3.1 The degree of an arbitrary vertex u of the bridge graph B 2 , d ≥ 2, is

given by:

⎧
⎨

deg G 1 ( u ) if u ∈ V ( G 1 ) −{ w 1 }

deg G d ( u ) if u ∈ V ( G d ) −{ v d }

deg B 2 ( u ) = ,

deg G i ( u ) if u ∈ V ( G i ) −{ v i , w i } ,2 ≤ i ≤ d − 1

⎩

ω i + 1 if u = w i ,1 ≤ i ≤ d − 1

υ i +

= ≤ ≤

1 if u v i ,2 i d

where υ i = deg G i ( v i ), ω i = deg G i ( w i ), for 1 ≤ i ≤ d ,

Theorem 7.3.2 The first Zagreb index of the bridge graph B 2 , d ≥ 2, is given by:

d d − 1 d

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

i = 1 i = 1 i = 2

where υ i = deg G i ( v i ), ω i =

≤ ≤

deg G i ( w i ), for 1 i d .

Proof Using the definition of the first Zagreb index, and Lemma 7.3.1, we have:

d − 1

deg G 1 ( u ) 2 deg G i ( u ) 2 M 1 ( B 2 ) = +

u ∈ V ( G 1 ) −{ w 1 } i = 2 u ∈ V ( G i ) −{ v i , w i }

d − 1 d

deg G d ( u ) 2 1) 2 1) 2 + + ( ω i + + ( υ i +

u ∈ V ( G d ) −{ v d } i = 1 i = 2

d − 1 d − 1 d − 1

− ω 1 2 υ i 2 ω i 2 − υ d 2

= M 1 ( G 1 ) + M 1 ( G i ) − − + M 1 ( G d )

i = 2 i = 2 i = 2

d − 1 d − 1 d d

ω i 2 υ i 2 + + 2 ω i + d − 1 + + 2 υ i + d − 1

i = 1 i = 1 i = 2 i = 2

d d − 1 d

= M 1 ( G i ) + 2 ω i + 2 υ i + 2 d − 2 .

i = 1 i = 1 i = 2