The First and Second Zagreb Indices of Several Interesting Classes of Chemical Graphs and Nanostructures - Exotic Properties of Carbon Nanomatter - page 153

Fig. 7.1 The bridge graph
B 1 = B 1 ( G 1 , G 2 , ... ,
G d ; v 1 , v 2 , ... , v d )

i

respect to the vertices

1 is the graph obtained from the graphs G 1 , G 2 , ... , G d
by connecting the vertices v i and v i + 1 by an edge for all i

{ v i }

=

∈{ − }

1,2, ... , d 1 ,as

shown in Fig. 7.1 .
First, we express the following simple lemma which is crucial in this section. Its
proof follows immediately from the definition of B 1 , so is omitted.
Lemma 7.2.1 The degree of an arbitrary vertex u of the bridge graph B 1 , d ≥

2, is

given by:

⎧
⎨

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

υ 1 + 1 if u = v 1

deg B 1 ( u ) = ,

⎩

υ i + 2 if u = v i ,2 ≤ i ≤ d − 1

υ d + 1 if u = v d

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

Theorem 7.2.2 The first Zagreb index of the bridge graph B 1 , d ≥ 2, is given by:

d d − 1

M 1 ( B 1 ) = M 1 ( G i ) + 2 υ 1 + 4 υ i + 2 υ d + 4 d − 6,

i = 1 i = 2

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

Proof Using the definition of the first Zagreb index, and Lemma 7.2.1, we have:

d d − 1

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

i = 1 u ∈ V ( G i ) −{ v i } i = 2

d d d − 1 d − 1

υ i 2 υ 1 2 υ i 2

= M 1 ( G i ) − + + 2 υ 1 + 1 + + 4 υ i

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

υ d 2

+ 4( d − 2) + + 2 υ d + 1

d d − 1

= M 1 ( G i ) + 2 υ 1 + 4 υ i + 2 υ d + 4 d − 6 .

i = 1 i = 2