The First and Second Zagreb Indices of Several Interesting Classes of Chemical Graphs and Nanostructures - Exotic Properties of Carbon Nanomatter - page 174

Fig. 7.20 The double starlike
tree S (5,3, 4,2, 2; 4,1, 2,3)

Let k 1 , k 2 , ... , k n be n positive integers. Set I 1 ={

| ≤ ≤

Corollary 7.5.13 i 1 i

n , k i = 1 } , I 2 ={ i | 1 ≤ i ≤ n , k i = 2 } and I 3 ={ i | 1 ≤ i ≤ n , k i ≥ 3 } and

let r . The first and second Zagreb indices of the starlike tree

S ( k 1 , k 2 , ... , k n ) are given by:
1. M 1 ( S ( k 1 , k 2 , ... , k n ))

| I 1 | = t , | I 2 | =

4 i ∈ I 2 I 3 k i , n 2 = − 3 n + 4 t +

4 i ∈ I 3 k i . 2 n 2

2. M 2 ( S ( k 1 , k 2 , ... , k n )) = − 6 n + 8 r − t ( n − 6) +

The starlike tree S ( k 1 , k 2 , ... , k n ) is said to be regular, if k 1 = k 2 = ... = k n = k .

Clearly S ( 1 ,1, .. . , 1 ) = S n + 1 . Using Corollary 7.5.13, we can get the following

ntimes

formulae for Zagreb indices of regular starlike trees.

Corollary 7.5.14 Let k be a positive integer. The first and second Zagreb indices of
the regular starlike tree S ( k , k , .. . , k ) are given by:

ntimes

n 2

1. M 1 ( S ( k , k , .. . , k )) = − 3 n + 4 nk ,

⎧
⎨

ntimes

n 2 if k = 1

2. M 2 ( S ( k , k , .. . , k )) = .

⎩

2 n 2 − 6 n + 4 nk if k ≥ 2

ntimes

Let S ( k 1 , k 2 , ... , k n ) denote the starlike tree which has a vertex v of degree greater
than two and has the property S ( k 1 , k 2 , ... , k n )

P k 1 P k 2 ... P k n . Also
let S ( k 1 , k 2 , ... , k n ) denote the starlike tree which has a vertex v of degree greater
than two and has the property S ( k 1 , k 2 , ... , k n )

− =

v

P k 1 P k 2 ... P k n .
The graph S ( k 1 , k 2 , ... , k n ; k 1 , k 2 , ... , k n ) obtained by joining the vertex v of the
graph S ( k 1 , k 2 , ... , k n ) to the vertex v of the graph S ( k 1 , k 2 , ... , k n ) by an edge is
called double starlike tree, see Fig. 7.20 . It has exactly two adjacent vertices v and
v of degree greater than three and has the property

v =

−