The First and Second Zagreb Indices of Several Interesting Classes of Chemical Graphs and Nanostructures - Exotic Properties of Carbon Nanomatter - page 175

v , v }= P k 1 P k 2 ... P k n P k 1
P k 2 ... P k n .

S ( k 1 , k 2 , ... , k n ; k 1 , k 2 , ... , k n ) −{

We can consider the double starlike tree S ( k 1 , k 2 , ... , k n ; k 1 , k 2 , ... , k n ), as the
bridge graph B 1 ( S ( k 1 , k 2 , ... , k n ), S ( k 1 , k 2 , ... , k n ); v , v ). So using Theorems
7.2.2, 7.2.4, and then Corollary 7.5.13, we can easily get the following result.

k 1 , k 2 , ... , k n }

Corollary 7.5.15 Let { k 1 , k 2 , ... , k n } and { be two sequences of

positive integers. Set I 1 ={ i | 1 ≤ i ≤ n , k i = 1 } , I 2 ={ i | 1 ≤ i ≤ n , k i = 2 } , I 3 =

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

r , I 1 = , and I 3 ={ n , k i ≥ t ,

2 } i | 1 ≤ i ≤ 3 } . Also let | I 1 | = t , | I 2 | =

and I 2 = r . The first and second Zagreb indices of the double starlike tree

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

4
i ∈ I 2 I 3

k i

i ∈ I 2 I 3

1. M 1 ( S ( k 1 , k 2 , ... , k n ; k 1 , k 2 , ... , k n ))

= k i + + n ( n − 1)

n ( n − t )

2,
2. M 2 ( S ( k 1 , k 2 , ... , k n ; k 1 , k 2 , ... , k n ))
=

+ 1) + 4( t + +

4 i ∈ I 3 k i + i ∈ I 3 k i + n (2 n − t −

n (2 n − t − 3) + 3)

r ) t ) nn + + 8( r + + 5( t + + 1 .

n and k i = k i for 1

If n n , then the double starlike tree

S ( k 1 , k 2 , ... , k n ; k 1 , k 2 , ... , k n ) is called symmetric double starlike. Using Corol-
lary 7.5.15, we get the following formulae for the first and second Zagreb indices of
symmetric double starlike trees.
Corollary 7.5.16 Let k 1 , k 2 , ... , k n be positive integers. Set I 1 ={

= ≤ i ≤

| ≤ ≤

n , k i = i 1 i

}

, I 2 ={

| ≤ ≤

n , k i =

}

, and I 3 ={

| ≤ ≤

n , k i ≥

} | I 1 | =

1 i 1 i 2 i 1 i 3 and let t ,

| I 2 | = r . The first and second Zagreb indices of the symmetric double starlike tree

S ( k 1 , k 2 , ... , k n ; k 1 , k 2 , ... , k n ) are given by:
1. M 1 ( S ( k 1 , k 2 , ... , k n ; k 1 , k 2 , ... , k n )) 8 i ∈ I 2 I 3 k i + = 2 n ( n − 1) + 8 t + 2,

8 i ∈ I 3 k i + 5 n 2

2. M 2 ( S ( k 1 , k 2 , ... , k n ; k 1 , k 2 , ... , k n )) = − 2 n ( t + 3) + 16 r

+ 10 t + 1 .

Now, we consider generalized Bethe trees. The level of a vertex in a rooted tree is one
more than its distance from the root vertex. A generalized Bethe tree of k levels, k> 1
is a rooted tree in which vertices at the same level have the same degree (Rojo 2007 ).
Let B k be a generalized Bethe tree of k levels. For i , we denote by

d k − i + 1 and n k − i + 1 the degree of the vertices at the level i of B k and their number,
respectively. Also, suppose e k =

∈{ 1,2, ... , k }

d k and e i = d i − 1 for i ∈{ 1,2, ... , k − 1 } . Thus,

d 1 = 1 is the degree of the vertices at the level k (pendent vertices) and d k is the

degree of the root vertex. On the other hand, n k = 1 is pertaining to the single vertex

at the first level, the root vertex. For i , suppose β k − i + 1 denotes the

subtree of B k which is also a generalized Bethe tree of k

∈{ 1, 2, ... , k − 1 }

1 levels and its root is
any vertex of the level i of B k , as shown in Fig. 7.21 and let β 1 =

− i +

K 1 . Now consider

∈{ − }

the subtree of β k − i + 1 , i 1,2, ... , k 1 , which is isomorphic to the star graph