**Int.** **J.** **IndustrialMathematics (ISSN 2008-5621)**
Vol. 9, No. 1, 2017 Article ID IJIM-00865, 11 pages

Research Article

### Zagreb, multiplicative Zagreb Indices and Coindices of graphs

V. Ahmadi *∗†*, M. R. Darafsheh *‡*, J. Hashemi *§*

Received Date: 2015-04-26 Revised Date: 2016-05-15 Accepted Date: 2016-05-25

**————————————————————————————————–**

**Abstract**

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The first, second and third
Zagreb indices of G are respectivly defined by: *M*1(*G*) =

∑

*u∈V* *d*(*u*)
2_{, M}

2(*G*) =
∑

*uv∈Ed*(*u*)*.d*(*v*)

and *M*3(*G*) =
∑

*uv∈E|d*(*u*)*−d*(*v*)*|* , where d(u) is the degree of vertex u in G and uv is an edge

of G connecting the vertices u and v. Recently, the first and second multiplicative Zagreb indices
of G are defined by: *P M*1(*G*) =

∏

*u∈V* *d*(*u*)

2 _{and} * _{P M}*
2(

*G*) =

∏

*u∈V* *d*(*u*)

*d*(*u*)_{. The first and second}

Zagreb coindices of G are defined by: *M*1(*G*) =
∑

*uv /∈E*(*d*(*u*) +*d*(*v*)) and*M*2(*G*) =
∑

*uv /∈Ed*(*u*)*.d*(*v*).

The indices*P M*1(*G*) =
∏

*uv /∈Ed*(*u*) +*d*(*v*) and*P M*2(*G*) =
∏

*uv /∈Ed*(*u*)*.d*(*v*) , are called the first and

second multiplicative Zagreb coindices of G, respectively. In this article, we compute the first, second and third Zagreb indices and the first and second multiplicative Zagreb indices of some classes of dendrimers. The first and second Zagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.Also, the multiplicative Zagreb indices are computed using link of graphs.

*Keywords* : Zagreb indices; Multiplicative Zagreb indices; Zagreb coindices; Multiplicative Zagreb
coindices; Link.

**—————————————————————————————————–**

**1**

**Introduction**

## T

hand connected. Let G=(V,E) be a simplee graphs considered in this paper are simple connected graph with vertex set V and edge set E. A topological index is a fixed number under graph automorphisms. Gutman and Trinajsti [4] defined the first and second Zagreb indices. Za-greb indices are defined as follows:*M*1(*G*) =

∑

*u∈V*

*d*(*u*)2*,* (1.1)

*∗*_{Corresponding author. vidaahmadi math@yahoo.com}
*†*_{Department of Mathematics, Shahid Chamran }

Uni-versity, Ahvaz, Iran.

*‡*_{Department} _{of} _{Mathematics,} _{Tehran} _{University,}

Tehran, Iran.

*§*_{Department of Mathematics, Shahid Chamran }

Uni-versity, Ahvaz, Iran.

*M*2(*G*) =

∑

*uv∈E*

*d*(*u*)*.d*(*v*) (1.2)

The alternative expression of*M*1(*G*) is

∑

*uv∈E*

(*d*(*u*) +*d*(*v*))*.* (1.3)

G.H.Fath-Tabar [3] defined the third Zagreb in-dex, by:

*M*3(*G*) =

∑

*uv∈E*

*|d*(*u*)*−d*(*v*)*|.* (1.4)

Todeschine et al. [6,7] have recently proposed to consider multiplicative variants of additive graph invariants, applied to the Zagreb indices, lead to:

*P M*1(*G*) =

∏

*u∈V*

*d*(*u*)2*,* (1.5)

*P M*2(*G*) =

∏

*u∈V*

*d*(*u*)*d*(*u*) (1.6)

The alternative expression of*P M*2(*G*) is

∏

*uv∈E*

*d*(*u*)*.d*(*v*)*.* (1.7)

Gutman et al. [5] computed multiplicative Za-greb indices for a class of dendrimers using link of graphs.

Recently, Ashrafi,Doli and Hamzeh [1, 2] de-fined the first and second Zagreb coindices by:

*M*1(*G*) =

∑

*uv /∈E*

(*d*(*u*) +*d*(*v*))*,* (1.8)

*M*2(*G*) =

∑

*uv /∈E*

*d*(*u*)*.d*(*v*)*.* (1.9)

In 2013 Xu, Das and Tang [8] defined multiplica-tive Zagreb coindices by:

*P M*1(*G*) =

∏

*uv /∈E*

*d*(*u*) +*d*(*v*)*,* (1.10)

*P M*2(*G*) =

∏

*uv /∈E*

*d*(*u*)*.d*(*v*)*.* (1.11)

They define multiplicative sum Zagreb index and the total multiplicative sum Zagreb index by:

*P M*1*∗*(*G*) =

∏

*uv∈E*

*d*(*u*) +*d*(*v*)*,* (1.12)

*P MT*(*G*) = ∏

*u,v∈V*

*d*(*u*) +*d*(*v*)*.* (1.13)

The goal of this article is computing Zagreb indices, multiplicative Zagreb indices, Zagreb coindices, multiplicative Zagreb coindices, mul-tiplicative sum Zagreb index and the total multi-plicative sum Zagreb index of some graphs.

**2**

**Preliminaries**

Define *di* to be the number of vertices with
de-grees i and *xij, i̸*=*j*, to be the number of edges
connecting the vertex of degree i with a vertex of
degree j and *xii* to be the number of edges
con-necting two vertices of degree i. Figure1shows a
first-type nanostar dendrimer which has grown
two stages and four similar branches with the
same number, ´*di* of vertices and ´*xij* of edges
con-necting a vertex of degree i with a vertex of degree
j and ´*xii*to be the number of edges connecting two
vertices of degree i. Define*xij* to be the number
of subsets with vertices of degree i, j, so that*xij*

does not include the number of edges that
con-nect vertices i, j. Define *xii* to be the number of
subsets with vertices of degree i, so that *xii*does

not include the number of edges which connect two vertices of degree i.

**Lemma 2.1** *The amount ofxij, xiiare equal to:*

*xij* =
(

*di*

1

) (
*dj*

1

)

*−xij* =*didj−xij,* (2.14)

*xii*=

(
*di*

2

)

*−xii*= *di*(*di−*1)

2 *−xii.* (2.15)

**Proof.** straight forward. □

We use the above formulae to obtain Zagreb and multiplicative Zagreb coindices.

**Lemma 2.2** *The number of subsets with *
*ver-tices of degree i as well as the number of subsets*
*with vertices of degree i, j, are equal to:*

(
*di*

2

)

= *di*(*di−*1)
2 *,*

(
*di*

1

) (
*dj*

1

)

=*didj.* (2.16)

**Proof.** straight forward. □We use these
formu-lae to obtain *P MT*(*G*).

**Definition 2.1** *A link of G and H by vertices*
*u and v is defined as the graph* (*G*□*H*)(*u, v*)
*ob-tained by joining u and v by an edge in the union*
*of these graphs.*

**Theorem 2.1** *The first and second *
*multiplica-tive Zagreb indices of the link of* *G*1 *and* *G*2
*sat-isfies the following relations:*

*PM˙1 (G˙1*□*G*2)(*v*1*, v*2) =

((*dG*1(*v*1)+1)(*dG*2(*v*2)+1)

*dG*1(*v*1)*dG*2(*v*2) )

2_{.}

*P M*1(*G*1)*P M*1(*G*2)*,* (2.17)

*PM˙2 (G˙1*□*G*2)(*v*1*, v*2) =
(*dG*_{1}(*v*1)+1)*dG*1

(*v*_{1)+1}

(*dG*_{2}(*v*2)+1)*dG*2
(*v*_{2)+1}

*dG*1(*v*1)*dG*1
(*v*1)_{d}

*G*2(*v*2)*dG*2

(*v*2) *.*

*P M*2(*G*1)*P M*2(*G*2)*.* (2.18)

**Definition 2.2** *Define* *P M*(*ui*)

1 (*G*1) *to be the*
*first multiplicative Zagreb index of the graph* *G*1,
*so thatP M*(*ui*)

1 (*G*1)*dose not include the first *
*mul-tiplicative Zagreb index of vertex* *ui* *∈V*(*G*1) *and*

*P M*(*ui*)

2 (*G*1) *is defined similarly.*

We compute these indices for the following fig-ures.

**3**

**Results and discussion**

**Theorem 3.1** *Zagreb, multiplicative Zagreb *
*in-dices and coinin-dices of First-type nanostar *
*den-drimer,* *N S*1[2] *(see Figure* *1) are computed as*
*follows:*

*M*1 = 17*.*2*n*+4*−*112*,*

*M*2 = 41*.*2*n*+3*−*140*,*

*M*3 = 2*n*+5*−*16*,*

*P M*1 = 22
*n*+6_{−}_{20}

*.*32*n*+5*−*16*,*
*P M*2 = 22

*n*+6_{−}_{20}

*.*33*.*2*n*+4*−*24*,*
*M*1 = 21*.*22*n*+8*−*141*.*2*n*+5+ 948*,*

*M*2 = 49*.*22*n*+7*−*337*.*2*n*+4+ 1164*,*

*P M*1 = 29*.*2

2*n*+7_{−}_{53}_{.}_{2}*n*+4_{+154}

*.*

322*n*+7*−*9*.*2*n*+4+40*.*522*n*+9*−*7*.*2*n*+6+96*,*
*P M*2 = 23*.*2

2*n*+9_{−}_{9}_{.}_{2}*n*+7_{+210}

*.*

33*.*22*n*+8*−*23*.*2*n*+5+176*.*

**Proof.** Figure1, has grown four similar branches
and two stages. Now, we compute these indices
from the stage n. For the graph of Figure 2 is
the central part of Figure 1. For the Figure 2
it is obvious that: *d*2 = 18*, d*3 = 12*,* so *d*2 =

4 ´*d*2+ 18*, d*3 = 4 ´*d*3+ 12*.*

For example, for *n* = 1 we have *d*2 = 54*, d*3 =

24. On the other hand, calculations show that: ´

*d*2 = 2*n*+3 *−*7*,d*´3 = 2*n*+2 *−*5*,* therefore, *d*2 =

2*n*+5*−*10*, d*3 = 2*n*+4*−*8*.* Elementary

com-putation gives:

*M*1 = 17*.*2*n*+4*−*112*,*

*P M*1 = 22
*n*+6_{−}_{20}

*.*32*n*+5*−*16*,*
*P M*2 = 22

*n*+6_{−}_{20}

*.*33*.*2*n*+4*−*24*.*

For the graph of Figure 2, it is obvious that:

*x*22 = 6*, x*23 = 24*, x*33 = 4*,* so *x*22 = 4 ´*x*22 +

6*, x*23 = 4 ´*x*23+ 24*, x*33 = 4 ´*x*33+ 4*.*For example,

for*n*= 1 we have x˙22=30, x˙23=48, x˙33=12.
On the other hand, calculations show that:
´

*x*22 = 2*n*+2*−*2*,x*´23= 2*n*+3*−*10*,x*´33 = 2*n*+1*−*2*,*

therefore,*x*22= 2*n*+4*−*2*, x*23= 2*n*+5*−*16*, x*33=

2*n*+3*−*4*.*

Elementary computation gives:

*M*2 = 41*.*2*n*+3*−*140*, M*3= 2*n*+5*−*16*.*

**Figure 1:** First-type nanostar dendrimer,*N S*1[2]

**Figure 2:** First-type nanostar dendrimer,*N S*1[2]

Similar calculation shows that:

*M*1 = 21*.*22*n*+8*−*141*.*2*n*+5+ 948*,*

*M*2 = 49*.*22*n*+7*−*337*.*2*n*+4+ 1164*,*

*P M*1 = 29*.*2

2*n*+7_{−}_{53}_{.}_{2}*n*+4_{+154}

*.*

322*n*+7*−*9*.*2*n*+4+40*.*522*n*+9*−*7*.*2*n*+6+96*,*
*P M*2 = 23*.*2

2*n*+9_{−}_{9}_{.}_{2}*n*+7_{+210}

*.*

33*.*22*n*+8*−*23*.*2*n*+5+176*.*

Now, we compute multiplicative Zagreb indices
using the link of graphs*G*1*, G*2as shown in Figure
3: It is easy to see that:

*P M*1(*G*1) = 220*.*34*,*

*P M*(*ui*)

1 (*G*1) = 218*.*34*,*

*P M*(*vj*)

1 (*G*1) = 218*.*34*,*

*P M*(*vi,ui*+1)

1 (*G*1) = 216*.*34

and so, for 1⩽*i*⩽*n−*1*,*1⩽*j*⩽*n−*1*.*

We define*Gnas* follows:
*Gn*= (*Gn−*1□*G*1)(*v*1*, u*1)

*Gn−*1 = (*Gn−*2□*G*1)(*v*2*, u*2)

. . .

**Figure 3:** *G*1and*G*2

**Figure 4:** Molecular graph of dendrimers using
tetrathiafulvalene units as branching centers, D [2]

According to Theorem 3.1 we have the follow-ing relations:

*P M*1(*Gn*) =

*P M*(*v*1)

1 (*Gn−*1)*P M*1(*u*1)(*G*1)*.*34

*P M*(*v*1)

1 (*Gn−*1) =

*P M*(*v*2)

1 (*Gn−*2)*P M*
(*v*1*,u*2)

1 (*G*1)*.*34

. . .

*P M*(*vn−*2)

1 (*G*2) =

*P M*(*vn−1*)

1 (*G*1)*P M*1(*vn−2,un−1*)(*G*1)*.*34

Therefore:

*P M*1(*Gn*) =*P M*_{1}(*vn−*1)(*G*1)*P M*_{1}(*u*1)(*G*1)

∏*n−*1

*i*=2 *P M*

(*vi−*1*,ui*)

1 (*G*1)*.*34(*n−*1) =

*P M*(*vn−1*)

1 (*G*1)*P M*1(*u*1)(*G*1)

(*P M*(*v*1*,u*2)

1 (*G*1))*n−*2*.*34(*n−*1)=

216*n*+4*.*38*n−*4*.*

A branch is added in the second layer such as

*G*1 and three branches are added in the third

layer that its first multiplicative Zagreb index is
2*−*2*.*218*.*318*.*The total number of added branches
in all layers are equal to 2*n−n−*1*.*

The first multiplicative Zagreb index for a main
branch of the graph of Figure 1 is equal to
22*n*+4*−*12*.*32*n*+3*−*12*.*

Since the graph has four main branches then we obtain high values for four main branches and

**Figure 5:** The central part of Figure4

**Figure 6:** *G*1 and*G*2

we consider the obtained number with the first
multiplicative Zagreb index which has the central
part value of figure that is 2*−*8*.*236*.*332*.*Therefore:

*P M*1(*G*) = 22
*n*+6_{−}_{20}

*.*32*n*+5*−*16_{.}

It is easy to see that:

*P M*2(*G*1) = 220*.*36*, P M*2(*ui*)(*G*1) = 218*.*36*,*

*P M*(*vj*)

2 (*G*1) = 218*.*36*, P M*

(*vi,ui*+1)

2 (*G*1) =

216*.*36

and so, for 1⩽*i*⩽*n−*1*,*1⩽*j*⩽*n−*1*.*

According to Theorem3.1 we have the follow-ing relations:

*P M*2(*Gn*) =

*P M*(*v*1)

2 (*Gn−*1)*P M*2(*u*1)(*G*1)*.*(33*.*33)

*P M*(*v*1)

2 (*G*1) =

*P M*(*v*2)

2 (*Gn−*2)*P M*
(*v*1*,u*2)

2 (*G*1)*.*(33*.*33) .

. .

*P M*(*vn−2*)

2 (*G*2) =

*P M*(*vn−1*)

2 (*G*1)*P M*2(*vn−2,un−1*)(*G*1)*.*(33*.*33)

Therefore:

*P M*2(*Gn*) =*P M*2(*vn−*1)(*G*1)*P M*2(*u*1)(*G*1)

(*P M*(*v*1*,u*2)

**Figure 7:** Polymer dendrimer, P [2]

**Figure 8:** The central part of Figure7

216*n*+4*.*312*n−*6*.*

The second multiplicative Zagreb index for
a main branch of the graph G is equal to
22*n*+4*−*12*.*33*.*2*n*+2*−*18*.*

Therefore,

*P M*2(*G*) = 22
*n*+6_{−}_{20}

*.*33*.*2*n*+4*−*24*.*□

**Theorem 3.2** *Zagreb, multiplicative Zagreb *
*in-dices and coinin-dices of Molecular graph of *
*den-drimers using tetrathiafulvalene units as *
*branch-ing centers, D [2] (see Figure* *4) are computed as*
*follows:*

*M*1 = 21*.*2*n*+5*−*414*,*

*M*2 = 99*.*2*n*+3*−*493*,*

*M*3 = 13*.*2*n*+3*−*64*,*

*P M*1 = 219*.*2
*n*+3_{−}_{88}

*.*35*.*2*n*+4*−*52*,*
*P M*2 = 219*.*2

*n*+3_{−}_{88}

*.*315*.*2*n*+3*−*78*,*

**Figure 9:** *G*1 and*G*2

**Figure 10:** Helicene-based dendrimers, H [2]

*M*1 = 1085*.*22*n*+5*−*5344*.*2*n*+3+ 13164*,*

*M*2 = 1225*.*22*n*+5*−*6091*.*2*n*+3+ 15150*,*

*P M*1 = 2453*.*2

2*n*+4_{−}_{271}_{.}_{2}*n*+5_{+2598}

*.*

311*.*22*n*+7*−*433*.*2*n*+2+536*.*

595*.*22*n*+5*−*957*.*2*n*+2+1200*,*
*P M*2 = 2589*.*2

2*n*+4_{−}_{2827}_{.}_{2}*n*+2_{+3388}

*.*

3155*.*22*n*+5*−*793*.*2*n*+3+2028*.*

**Proof.** Figure 4, has grown four similar
branches and two stages. Now, we compute these
indices from the stage n. Figure 5 is the
cen-tral part of Figure 4. For the graph of
Fig-ure 5 it is obvious that: *d*2 = 36*, d*3 = 14 so

*d*2= 4 ´*d*2+ 36*, d*3= 4 ´*d*3+ 14*.*

For example, for*n* = 1 we have *d*1 = 12*, d*2 =

108*, d*3 = 54*.* On the other hand, calculations

**Figure 11:** The central part of Figure10

**Figure 12:** *G*1 and*G*2

5*.*2*n*+1 *−* 10*,* therefore, *d*1 = 2*n*+3 *−* 4*, d*2 =

19*.*2*n*+2*−*44*, d*3 = 5*.*2*n*+3*−*26*.*Elementary

com-putation gives:

*M*1 = 21*.*2*n*+5*−*414*,*

*P M*1 = 219*.*2
*n*+3_{−}_{88}

*.*35*.*2*n*+4*−*52*,*
*P M*2 = 219*.*2

*n*+3_{−}_{88}

*.*315*.*2*n*+3*−*78*.*

For the graph of Figure 5, it is obvious that:

*x*22 = 16*, x*23 = 36*, x*33 = 3*,* so *x*22 = 4 ´*x*22 +

16*, x*23 = 4 ´*x*23+ 36*, x*33 = 4 ´*x*33+ 3*.* For

exam-ple, for *n* = 1 we have *x*13 = 4*, x*12 = 8*, x*22 =

40*, x*23= 128*, x*33= 15*.*

On the other hand, calculations show
that: *x*´13 = 2*n* *−* 1*,x*´12 = 2*n,x*´22 =

7*.*2*n* *−* 8*,x*´23 = 23*.*2*n* *−* 23*,x*´33 = 3*.*2*n* *−* 3*,*

therefore, *x*13 = 2*n*+2 *−*4*, x*12 = 2*n*+2*, x*22 =

7*.*2*n*+2_{−}_{16}_{, x}

23= 23*.*2*n*+2*−*56*, x*33= 3*.*2*n*+2*−*9*.*

Elementary computation gives:

*M*2 = 99*.*2*n*+3*−*493*,*

*M*3 = 13*.*2*n*+3*−*64*.*

Similar calculation shows that:

**Figure 13:** Tetrathiafulvalene [*T T F*]21-glycol

dendrimer, T [3]

**Figure 14:** The central part of Figure14

*M*1 = 1085*.*22*n*+5*−*5344*.*2*n*+3+ 13164*,*

*M*2 = 1225*.*22*n*+5*−*6091*.*2*n*+3+ 15150*,*

*P M*1 = 2453*.*2

2*n*+4_{−}_{271}_{.}_{2}*n*+5_{+2598}

*.*

311*.*22*n*+7*−*433*.*2*n*+2+536*.*

595*.*22*n*+5*−*957*.*2*n*+2+1200*,*
*P M*2 = 2589*.*2

2*n*+4_{−}_{2827}_{.}_{2}*n*+2_{+3388}

*.*

3155*.*22*n*+5*−*793*.*2*n*+3+2028*.*

Now, we compute multiplicative Zagreb
in-dices using the link of graphs *G*1*, G*2 as shown in

Figure 6:

It is easy to see that:

*P M*1(*G*1) = 238*.*318*,*

*P M*(*ui*)

1 (*G*1) = 238*.*318*,*

*P M*(*vj*)

1 (*G*1) = 236*.*318*,*

*P M*(*vi,ui*+1)

1 (*G*1) = 236*.*318

and so, for 1⩽*i*⩽*n−*1*,*1⩽*j*⩽*n−*1*.*

Therefore:

**Figure 15:** *G*1 and*G*2

(*P M*(*v*1*,u*2)

1 (*G*1))*n−*2*.*62(*n−*1)=

238*n.*320*n−*2*.*

A branch is added in the second layer such as

*G*1 and three branches are added in the third

layer that its first multiplicative Zagreb index is
238*.*320*.* The total number of added branches in
all layers are equal to 2*n−n−*1*.*

The first multiplicative Zagreb index for a main
branch of the graph of Figure 4 is equal to
238*.*2*n−*38*.*320*.*2*n−*22*.*

Since the graph has four main branches then
we obtain high values for four main branches and
we consider the obtained number with the first
multiplicative Zagreb index which has the central
part value of figure that is 2*−*8*.*272*.*336*.*

Therefore:

*P M*1(*G*) = 2152*.*2
*n _{−}*

_{88}

*.*380*.*2*n−*52*.*

It is easy to see that:

*P M*2(*G*1) = 238*.*327*,*

*P M*(*ui*)

2 (*G*1) = 238*.*327*,*

*P M*(*vj*)

2 (*G*1) = 236*.*327*,*

*P M*(*vi,ui*+1)

2 (*G*1) = 236*.*327

and so, for 1⩽*i*⩽*n−*1*,*1⩽*j*⩽*n−*1*.*

Therefore:

*P M*2(*Gn*) =*P M*_{2}(*vn−*1)(*G*1)*P M*_{2}(*u*1)(*G*1)

(*P M*(*v*1*,u*2)

2 (*G*1))*n−*2*.*(22*.*33)*n−*1 =

238*n.*330*n−*3*.*

The second multiplicative Zagreb index for
a main branch of the graph G is equal to
238*.*2*n−*38*.*330*.*2*n−*33*.*

Therefore,

*P M*2(*G*) = 2152*.*2
*n _{−}*

_{88}

*.*3120*.*2*n−*78*.*□

**Theorem 3.3** *Zagreb, multiplicative Zagreb *
*in-dices and coinin-dices of Molecular graph of *

*Poly-mer dendriPoly-mer, P[2] (see Figure7) are computed*
*as follows:*

*M*1 = 159*.*2*n*+1*−*186*,*

*M*2 = 183*.*2*n*+1*−*204*,*

*M*3 = 18*.*2*n*+1*−*18*,*

*P M*1 = 245*.*2
*n*+1_{−}_{66}

*.*315*.*2*n*+1*−*12*,*
*P M*2 = 245*.*2

*n*+1_{−}_{66}

*.*345*.*2*n−*18_{,}

*M*1 = 8694*.*22*n−*11130*.*2*n*+ 3546*,*

*M*2 = 4761*.*22*n*+1*−*12117*.*2*n*+ 3825*,*

*P M*1 = 2279*.*2

2*n*+3_{−}_{1605}_{.}_{2}*n*+1_{+1191}

*.*

3495*.*22*n−1−*411*.*2*n−1*+21*.*5675*.*22*n−*399*.*2*n*+1+216*,*
*P M*2 = 22835*.*2

2*n _{−}*

_{3969}

_{.}_{2}

*n*

_{+1386}

*.*

3945*.*22*n−*1023*.*2*n*+258*.*

**Proof.** Figure 7, has grown three similar
branches and two stages. Now, we compute these
indices from the stage n. Figure8the central part
of Figure 7. For the graph of Figure8 it is
obvi-ous that: *d*2 = 15*, d*3 = 9*,*so*d*2= 3 ´*d*2+ 15*, d*3 =

3 ´*d*3+ 9*.*

For example, for *n* = 1 we have

*d*1 = 6*, d*2 = 57*, d*3 = 24*.* On the

other hand, calculations show that: ´

*d*1= 2*n,d*´2 = 15*.*2*n−*16*,d*´3 = 5*.*2*n−*5*,*therefore,

*d*1= 3*.*2*n, d*2= 45*.*2*n−*33*, d*3 = 15*.*2*n−*6*.*

Elementary computation gives:

*M*1 = 159*.*2*n*+1*−*186*,*

*P M*1 = 245*.*2
*n*+1_{−}_{66}

*.*315*.*2*n*+1*−*12*,*
*P M*2 = 245*.*2

*n*+1_{−}_{66}

*.*345*.*2*n−*18*.*

For the graph of Figure 8, it is obvious that:

*x*22 = 6*, x*23 = 18*, x*33 = 3*.* so *x*22 = 3 ´*x*22 +

6*, x*23 = 3 ´*x*23 + 18*, x*33 = 3 ´*x*33+ 3*.* For

exam-ple, for *n* = 1 we have *x*12 = 6*, x*22 = 30*, x*23 =

48*, x*33= 12*.*

On the other hand, calculations show that: ´

*x*12= 2*n,x*´22= 9*.*2*n−*10*,x*´23= 11*.*2*n−*12*,x*´33=

2*n*+1*−*1*.*

therefore, *x*12 = 3*.*2*n, x*22 = 27*.*2*n−*24*, x*23 =

33*.*2*n−*18*, x*33= 3*.*2*n*+1*.*

Elementary computation gives:

*M*2 = 183*.*2*n*+1*−*204*,*

*M*3 = 18*.*2*n*+1*−*18*.*

Similar calculation shows that:

*M*1 = 8694*.*22*n−*11130*.*2*n*+ 3546*,*

*M*2 = 4761*.*22*n*+1*−*12117*.*2*n*+ 3825*,*

*P M*1 = 2279*.*2

2*n*+3_{−}_{1605}_{.}_{2}*n*+1_{+1191}

*.*

3495*.*22*n−*1*−*411*.*2*n−*1+21*.*5675*.*22*n−*399*.*2*n*+1+216*,*
*P M*2 = 22835*.*2

2*n _{−}*

_{3969}

_{.}_{2}

*n*

_{+1386}

*.*

Now, we compute multiplicative Zagreb indices
using the link of graphs*G*1*, G*2as shown in Figure
9:

It is easy to see that:

*P M*1(*G*1) = 230*.*38*,*

*P M*(*ui*)

1 (*G*1) = 230*.*38*,*

*P M*_{1}(*vj*)(*G*1) = 228*.*38*,*

*P M*(*vi,ui*+1)

1 (*G*1) = 228*.*38

and so, for 1⩽*i*⩽*n−*1*,*1⩽*j*⩽*n−*1*.*

Therefore:

*P M*1(*Gn*) =*P M*_{1}(*vn−*1)(*G*1)*P M*_{1}(*u*1)(*G*1)

(*P M*(*v*1*,u*2)

1 (*G*1))*n−*2*.*62(*n−*1)=

230*n.*310*n−*2*.*

A branch is added in the second layer such as

*G*1 and three branches are added in the third

layer that its first multiplicative Zagreb index is
230_{.}_{3}10_{.}_{The total number of added branches in}

all layers are equal to 2*n−n−*1*.*

The first multiplicative Zagreb index for a main
branch of the graph of Figure 7 is equal to
230*.*2*n−*30*.*310*.*2*n−*12*.*

Since the graph has three main branches then
we obtain high values for three main branches
and we consider the obtained number with the
first multiplicative Zagreb index which has the
central part value of figure that is 2*−*6*.*230*.*324.
Therefore:

*P M*1(*G*) = 290*.*2
*n _{−}*

_{66}

*.*330*.*2*n−*12*.*

It is easy to see that:

*P M*2(*G*1) = 230*.*312*,*

*P M*(*ui*)

2 (*G*1) = 230*.*312*,*

*P M*(*vj*)

2 (*G*1) = 228*.*312*,*

*P M*(*vi,ui*+1)

2 (*G*1) = 228*.*312

and so, for 1⩽*i*⩽*n−*1*,*1⩽*j*⩽*n−*1*.*

Therefore:

*P M*2(*Gn*) =*P M*2(*vn−*1)(*G*1)*P M*2(*u*1)(*G*1)

(*P M*(*v*1*,u*2)

2 (*G*1))*n−*2*.*(22*.*33)*n−*1 =

230*n.*315*n−*3*.*

The second multiplicative Zagreb index for
a main branch of the graph G is equal to
230*.*2*n−*30*.*315*.*2*n−*18*.*

*P M*2(*G*) = 290*.*2
*n _{−}*

_{66}

*.*345*.*2*n−*18*.*□

**Theorem 3.4** *Zagreb, multiplicative Zagreb *
*in-dices and coinin-dices of Helicene-based dendrimers,*
*H[2] (see Figure* *10) are computed as follows:*

*M*1 = 43*.*2*n*+3*−*168*,*

*M*2 = 225*.*2*n*+1*−*226*,*

*M*3 = 7*.*2*n*+2*−*14*,*

*P M*1 = 211*.*2
*n*+2_{−}_{16}

*.*37*.*2*n*+3*−*30_{,}

*P M*2 = 211*.*2
*n*+2_{−}_{16}

*.*321*.*2*n*+2*−*45*,*
*M*1 = 891*.*22*n*+3*−*874*.*2*n*+3+ 1718*,*

*M*2 = 1089*.*22*n*+3*−*4403*.*2*n*+1+ 2232*,*

*P M*1 = 2277*.*2

2*n*+2_{−}_{263}_{.}_{2}*n*+2_{+249}

*.*

315*.*22*n*+5*−*259*.*2*n*+1+143*.*

577*.*22*n*+3*−*287*.*2*n*+1+131*,*
*P M*2 = 2297*.*2

2*n*+2_{−}_{513}_{.}_{2}*n*+1_{+216}

*.*

3189*.*22*n*+3*−*797*.*2*n*+1+420*.*

**Proof.** Figure 10, has grown two similar
branches and two stages. Now, we compute these
indices from the stage n. Figure11 is the central
part of Figure 10. For the graph of Figure 11
it is obvious that: *d*1 = 3*, d*2 = 12*, d*3 = 15 so

*d*1= 2 ´*d*1+ 3*, d*2 = 2 ´*d*2+ 12*, d*3 = 2 ´*d*3+ 15.

For example, for *n* = 1 we have

*d*1 = 7*, d*2 = 36*, d*3 = 41*.*On the other hand,

cal-culations show that: *d*´1 = 2*n*+1 *−* 2*,d*´2 =

11*.*2*n* *−* 10*,d*´3 = 7*.*2*n*+1 *−* 15*,* therefore,

*d*1= 2*n*+2*−*1*, d*2 = 11*.*2*n*+1*−*8*, d*3 = 7*.*2*n*+2*−*15*.*

Elementary computation gives:

*M*1 = 43*.*2*n*+3*−*168*,*

*P M*1 = 211*.*2
*n*+2_{−}_{16}

*.*37*.*2*n*+3*−*30*,*
*P M*2 = 211*.*2

*n*+2_{−}_{16}

*.*321*.*2*n*+2*−*45*.*

For the graph of Figure 11, it is obvious that:

*x*12 = 1*, x*13 = 2*, x*22 = 5*, x*23 = 13*, x*33 = 14

so *x*12 = 2 ´*x*12+ 1*, x*13 = 2 ´*x*13+ 2*, x*22 = 2 ´*x*22+

5*, x*23 = 2 ´*x*23+ 13*, x*33 = 2 ´*x*33+ 14*.* For

exam-ple, for *n* = 1 we have *x*12 = 1*, x*13 = 6*, x*22 =

21*, x*23= 29*, x*33= 44*.*

On the other hand, calculations show that: ´

*x*13 = 2*n*+1*−*2*,x*´22 = *x*´22 = 3*.*2*n*+1 *−*4*,x*´23 =

5*.*2*n*+1 *−*12*,x*´33 = 15*.*2*n−*15*,* therefore *x*12 =

1*, x*13= 2*n*+2*−*2*, x*22= 3*.*2*n*+2*−*3*, x*23= 5*.*2*n*+2*−*

11*, x*33= 15*.*2*n*+1*−*16*.*

Elementary computation gives:

*M*2 = 225*.*2*n*+1*−*226*,*

*M*3 = 7*.*2*n*+2*−*14*.*

Similar calculation shows that:

*M*1 = 891*.*22*n*+3*−*874*.*2*n*+3+ 1718*,*

*M*2 = 1089*.*22*n*+3*−*4403*.*2*n*+1+ 2232*,*

*P M*1 = 2277*.*2

2*n*+2_{−}_{263}_{.}_{2}*n*+2_{+249}

*.*

315*.*22*n*+5*−*259*.*2*n*+1+143*.*

577*.*22*n*+3*−*287*.*2*n*+1+131*,*
*P M*2 = 2297*.*2

2*n*+2_{−}_{513}_{.}_{2}*n*+1_{+216}

*.*

3189*.*22*n*+3*−*797*.*2*n*+1+420*.*

Now, we compute multiplicative Zagreb
in-dices using the link of graphs *G*1*, G*2 as shown in

It is easy to see that:

*P M*1(*G*1) = 14*.*226*.*324*,*

*P M*(*ui*)

1 (*G*1) = 14*.*224*.*324*,*

*P M*(*vj*)

1 (*G*1) = 14*.*224*.*324*,*

*P M*(*vi,ui*+1)

1 (*G*1) = 14*.*222*.*324

and so, for 1⩽*i*⩽*n−*1*,*1⩽*j*⩽*n−*1*.*

Therefore:

*P M*1(*Gn*) =*P M*_{1}(*vn−*1)(*G*1)*P M*_{1}(*u*1)(*G*1)

(*P M*(*v*1*,u*2)

1 (*G*1))*n−*2*.*34(*n−*1)=

222*n*+4_{.}_{3}28*n−*4_{.}

A branch is added in the second layer such as

*G*1 and three branches are added in the third

layer that its first multiplicative Zagreb index is
222*.*328*.* The total number of added branches in
all layers are equal to 2*n−n−*1*.*

The first multiplicative Zagreb index for a main
branch of the graph of Figure 10 is equal to
211*.*2*n*+1*−*18_{.}_{3}7*.*2*n*+2*−*32_{.}

Since the graph has two main branches then
we obtain high values for two main branches and
we consider the obtained number with the first
multiplicative Zagreb index which has the central
part value of figure that is 2*−*4*.*224*.*334*.*Therefore:

*P M*1(*G*) = 211*.*2
*n*+2_{−}_{16}

*.*37*.*2*n*+3*−*30*.*

It is easy to see that:

*P M*2(*G*1) = 12*.*226*.*336*,*

*P M*(*ui*)

2 (*G*1) = 12*.*224*.*336*,*

*P M*(*vj*)

2 (*G*1) = 12*.*224*.*336*,*

*P M*_{2}(*vi,ui*+1))(*G*1) = 12*.*222*.*336

and so, for 1⩽*i*⩽*n−*1*,*1⩽*j*⩽*n−*1*.*

Therefore:

*P M*2(*Gn*) =*P M*2(*vn−*1)(*G*1)*P M*2(*u*1)(*G*1)

(*P M*(*v*1*,u*2)

2 (*G*1))

*n−*2

*.*36(*n−*1)=
222*n*+4*.*342*n−*6*.*

The second multiplicative Zagreb index for
a main branch of the graph G is equal to
211*.*2*n−*18*.*321*.*2*n*+1*−*48*.*

*P M*2(*G*) = 211*.*2
*n*+1_{−}_{16}

*.*321*.*2*n*+2*−*45_{.}_{□}

**Theorem 3.5** *Zagreb,* *multiplicative* *Zagreb*
*indices* *and* *coindices* *of* *Tetrathiafulvalene*

[*T T F*]21-glycol Dendrimer (see Figure *13) are*
*computed as follows:*

*M*1 = 63*.*2*n*+2*−*204*,*

*M*2 = 303*.*2*n−*243*,*

*M*3 = 21*.*2*n*+1*−*30*,*

*P M*1 = 221*.*2
*n*+1_{−}_{48}

*.*39*.*2*n*+2*−*24*,*
*P M*2 = 221*.*2

*n*+1_{−}_{48}

*.*327*.*2*n*+1*−*36*,*

*M*1 = 2295*.*22*n*+1*−*3903*.*2*n*+1+ 3312*,*

*M*2 = 2601*.*22*n*+1*−*8997*.*2*n*+ 3873*,*

*P M*1 = 2837*.*2

2*n _{−}*

_{717}

_{.}_{2}

*n*+1

_{+699}

*.*

39*.*22*n*+5*−*189*.*2*n*+1+81*.*

5189*.*22*n*+1*−*357*.*2*n*+1+318*,*
*P M*2 = 2945*.*2

2*n _{−}*

_{1899}

_{.}_{2}

*n*

_{+936}

*.*

3405*.*22*n*+1*−*315*.*2*n*+2+480_{.}

**Proof.** Figure 13, has grown three similar
branches and three stages. Now, we compute
these indices from the stage n. Figure 14 is the
central part of Figure 13. For the graph of
Fig-ure 14 it is obvious that: *d*1 = 3*, d*3 = 6*,* so

*d*1= 3 ´*d*1+ 3*, d*3 = 2 ´*d*3+ 6*.*

For example, for *n* = 1 we have

*d*1 = 12*, d*2 = 18*, d*3 = 24*.* On the

other hand, calculations show that: ´

*d*1= 4*.*2*n−*1*,d*´2 = 7*.*2*n−*8*,d*´3= 6*.*2*n−*6*,*

there-fore, *d*1= 6*.*2*n, d*2= 21*.*2*n−*24*, d*3 = 18*.*2*n−*12*.*

Elementary computation gives:

*M*1 = 63*.*2*n*+2*−*204*,*

*P M*1 = 221*.*2
*n*+1_{−}_{48}

*.*39*.*2*n*+2*−*24*,*
*P M*2 = 221*.*2

*n*+1_{−}_{48}

*.*327*.*2*n*+1*−*36*.*

For the graph of Figure 14, it is obvious that:

*x*13= 3*, x*33= 6*.*so*x*13= 3 ´*x*13+3*, x*33= 3 ´*x*33+6*.*

For example, for *n* = 1 we have *x*22 = 3*, x*23 =

30*, x*13= 12*, x*33= 15*.*

On the other hand, calculations show that: ´

*x*22 = 2*n*+1*−*3*,x*´23 = 5*.*2*n*+1*−*10*,x*´13 = 2*n*+1*−*

1*,x*´33 = 3*.*2*n* *−* 3*,* therefore, *x*22 = 3*.*2*n*+1 *−*

9*, x*23= 15*.*2*n*+1*−*30*, x*13= 3*.*2*n*+1*, x*33= 9*.*2*n−*3*.*

Elementary computation gives:

*M*2 = 303*.*2*n−*243*,*

*M*3 = 21*.*2*n*+1*−*30*.*

Similar calculation shows that:

*M*1 = 2295*.*22*n*+1*−*3903*.*2*n*+1+ 3312*,*

*M*2 = 2601*.*22*n*+1*−*8997*.*2*n*+ 3873*,*

*P M*1 = 2837*.*2

2*n _{−}*

_{717}

_{.}_{2}

*n*+1

_{+699}

*.*

39*.*22*n*+5*−*189*.*2*n*+1+81_{.}

5189*.*22*n*+1*−*357*.*2*n*+1+318*,*
*P M*2 = 2945*.*2

2*n _{−}*

_{1899}

_{.}_{2}

*n*

_{+936}

*.*

3405*.*22*n*+1*−*315*.*2*n*+2+480*.*

Now, we compute multiplicative Zagreb
in-dices using the link of graphs *G*1*, G*2 as shown in

Figure 15:

It is easy to see that:

*P M*1(*G*1) = 210*.*312*,*

*P M*(*ui*)

1 (*G*1) = 210*.*312*,*

*P M*(*vj*)

1 (*G*1) = 210*.*312*,*

*P M*(*vi,ui*+1)

and so, for 1⩽*i*⩽*n−*1*,*1⩽*j*⩽*n−*1*.*

Therefore:

*P M*1(*Gn*) =*P M*1(*vn−1*)(*G*1)*P M*1(*u*1)(*G*1)

(*P M*(*v*1*,u*2)

1 (*G*1))*n−*2*.*24(*n−*1)=

214*n−*4*.*312*n.*

A branch is added in the second layer such as

*G*1 and three branches are added in the third

layer that its first multiplicative Zagreb index is
214*.*312*.* The total number of added branches in
all layers are equal to 2*n−n−*1*.*

The first multiplicative Zagreb index for a main
branch of the graph of Figure 13 is equal to
214*.*2*n−*18*.*312*.*2*n−*12*.*

Since the graph has three main branches then
we obtain high values for three main branches and
we consider the obtained number with the first
multiplicative Zagreb index which has the central
part value of figure that is 26*.*312*.*Therefore:

*P M*1(*G*) = 242*.*2
*n _{−}*

_{48}

*.*336*.*2*n−*24*.*

It is easy to see that:

*P M*2(*G*1) = 210*.*318*,*

*P M*(*ui*)

2 (*G*1) = 210*.*318*,*

*P M*(*vj*)

2 (*G*1) = 210*.*318*,*

*P M*(*vi,ui*+1)

2 (*G*1) = 210*.*318

and so, for 1⩽*i*⩽*n−*1*,*1⩽*j*⩽*n−*1*.*

Therefore:

*P M*2(*Gn*) =*P M*

(*vn−*1)

2 (*G*1)*P M*
(*u*1)

2 (*G*1)

(*P M*(*v*1*,u*2)

2 (*G*1))*n−*2*.*24(*n−*1)=

214*n−*4*.*318*n.*

The second multiplicative Zagreb index for
a main branch of the graph G is equal to
214*.*2*n−*18_{.}_{3}18*.*2*n−*18_{.}

*P M*2(*G*) = 242*.*2
*n _{−}*

_{48}

*.*354*.*2*n−*36*.*□

**References**

[1] A. R. Ashrafi, T. Doili, A. Hamzeh, *The *
*Za-greb coindices of graph operations*, Discrete
Applied Mathematics 158 (2010) 1571-1578.

[2] A. R. Ashrafi, T. Doli, A. Hamzeh,*Extremal*
*graphs with respect to the Zagreb coindices*,
MATCH Communications in Mathematical
and in Computer Chemistry 65 (2011)
85-92.

[3] G. H. Fath-Tabar, *Old and new Zagreb *
*in-dices of graphs*, MATCH Communications in
Mathematical and in Computer Chemistry
65 (2011) 79-84.

[4] I. Gutman, N. Trinajsti, *Graph theory and*
*molecular orbitals. III. Totalπ _{−}electron*

*en-ergy of alternant hydrocarbons*, Chemical Physics Letters 17 (1972) 535-538.

[5] A. Iranmanesh, M. A. Hosseinzadeh, I.
Gutman, *On multiplicative Zagreb indices*
*of graphs*, Iranian journal of mathematical
Chemistry 3 (2012) 145-154.

[6] R. Todeschini, D. Ballabio, V. Consonni,

*Novel molecular descriptors based on *
*func-tions of new vertex degrees. in:* *I. *
*Gut-man and B. Furtula (Eds.)*, Novel Molecular
Structure Descriptors-Theory and
Applica-tions I, University Kragujevac, Kragujevac
(2010) 73-100.

[7] R. Todeschini, V. Consonni,*New local vertex*
*invariants and molecular descriptors based*
*on functions of the vertex degrees*, MATCH
Communications in Mathematical and in
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[8] K. Xu, K. Ch. Das, K. Tang, *On the *
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Vida Ahmadi was born in Masjed-soleyman in 1974. She is a tenured lecturer at Mathematics Depart-ment of Abadan Islamic Azad University, Abadan, Iran. Ms. Ahmadi graduated from Shahid-Chamran University of Ahvaz, Ah-vaz, Iran, where she got her BSs, MSs, and Ph.D. in Mathematics. She has published two books and two articles in Iranian journals and one of her articles has been published in a foreign jour-nal out of Iran.

all his 37 years of teaching, Dr. Derafshe gained enormous honors and awards including Abuds Salam Award from Theoretical Physics and Tri-este Mathematics, Italy and Kharazmi Interna-tional Award, the 10thKharazmi InternaInterna-tional Award. Dr. Derafshe has run three terms as the member of the Executive Council of The Iranian Mathematical Society and is also the member of both The American and The London Mathemat-ical Societies. He has compiled and translated eight books which have been published. It is worth mentioning that Dr. Derafshe has super-vised 88 Ph.D. candidates dissertations as their advisors and has been elected as the Researcher of the Year in Tehran University in 2008 and 2011 as well as the Professor of the Year in Tehran Uni-versity in 2003.