The First and Second Zagreb Indices of Several Interesting Classes of Chemical Graphs and Nanostructures - Exotic Properties of Carbon Nanomatter - page 160

= deg G i ( a )deg G i ( b ) + deg G i ( a )(deg G i ( v i ) + 1)

i ∈ I ab ∈ E ( G i ); a , b = v i , w i ab ∈ E ( G i ); a ∈ V ( G i ) −{ w i } , b = v i

1) + deg G i ( a )(deg G i ( w i ) + 1) + (deg G i ( v i ) + 1)(deg G i ( w i ) +

ab ∈ E ( G i ); a ∈ V ( G i ) −{ v i } , b = w i

= + + +

( M 2 ( G i ) α G i ( v i ) α G i ( w i ) 1) .

i ∈ I

Suppose I ={ i | 2 ≤ i ≤ d − 1, v i w i / ∈ E ( G i ) }={ 2,3, ... , d − 1 }− I . The forth

∈ I . Using Lemma 7.3.1, we

sum S 4 is taken over all edges ab ∈ E ( G i ) for all i

have:

S 4 = deg B 2 ( a )deg B 2 ( b )

i ∈ I

ab ∈ E ( G i )

= deg G i ( a )deg G i ( b ) + deg G i ( a )[deg G i ( v i ) + 1]

i ∈ I

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

1) + deg G i ( a )(deg G i ( w i ) +

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

= ( M 2 ( G i ) + α G i ( v i ) + α G i ( w i )) .

∈ I

i

Finally, the last sum S 5 is taken over all edges w i v i + 1 ,1 ≤ i ≤ d − 1. Using Lemma

7.3.1, we have:

d − 1 d − 1

S 5 = deg B 2 ( a )deg B 2 ( b ) = (deg G i ( w i ) + 1)(deg G i + 1 ( v i + 1 ) + 1)

i = 1 ab = w i v i + 1 i = 1

d − 1 d d − 1

= ω i + υ i + ω i υ i + 1 + d − 1 .

i = 1 i = 2 i = 1

Adding the quantities S 1 , S 2 , S 3 , S 4 , S 5 , we can get the formula for M 2 ( B 2 ) .

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 . Using Theorem 7.3.4 yields:

Corollary 7.3.5 Let υ = deg G ( v ) and ω = deg G ( w ). If v and w are adjacent in G ,

then

M 2 ( B 2 ) = dM 2 ( G ) + ( d − 1)[ υ + ω + υω + α G ( v ) + α G ( w ) + 2] − 1,

whereas otherwise;

M 2 ( B 2 ) = dM 2 ( G ) + ( d − 1)[ υ + ω + υω + α G ( v ) + α G ( w ) + 1] .

Now, we apply our results to compute the first and second Zagreb indices of some
chemical graphs and nano-structures.