**_______________________________________________________________________________________________**

**9 **

## On Some Multiplicative Topological Indices

M. Bhanumathi

Associate professor, PG Research of Mathematics, Govt. Arts College for Women(Autonomous),Pudukottai-622001,India

*bhanu_ksp@yahoo.com *

K. Easu Julia Rani

Assistant Professor of Mathematics, TRP Engineering College,Trichy-621105,India

*juliarani16@gmail.com*

* Abstract— The topological indices correlate certain physicochemical properties such as boiling point, stability of chemical *
compounds. In this paper, we compute the first and second Multiplicative Zagreb indices , the first and second multiplicative
hyper Zagreb indices, first and second Multiplicative modified Zagreb indices, the multiplicative First and second Zagreb
polynomial, multiplicative ABC index, multiplicative GA index, multiplicative Fifth GA index, multiplicative Fourth ABC
index and multiplicative Augmented Zagreb index for some special graphs.

**Keywords-First and second multiplicative Zagreb indices, the first and second multiplicative hyper Zagreb indices, first and second ***multiplicative modified Zagreb indices, multiplicative First and second Zagreb polynomial, multiplicative ABC index, multiplicative GA index, *
*multiplicative Fifth GA index, multiplicative Fourth ABC index and multiplicative Augmented Zagreb index *

__________________________________________________*****_________________________________________________

**I. ** **I****NTRODUCTION **

Writing the mathematical model of a problem in various sciences is a helpful tool which helps a great deal to their progress. As an example, a graph can be drawn in chemistry based on atoms and the existing bonds between them for every molecule and the graph mathematical models can be defined in order to analyze the molecule. Topological indices are one of the mathematical models that can be defined by assigning a real number to the chemical molecule. The physical-chemical characteristics of the molecules can be analyzed by taking benefit from the topological indices and such properties as boiling point, entropy, enthalpy of vaporization, standard enthalpy of vaporization, enthalpy of formation, Acetic factor, etc can be predicted.

Consider the simple graph G, with the set of the following vertices V = v1,v2,v3,… vn . If u ∈ V, therefore the number of the edges ending in is defined as the degree of vertex u and is denoted by deg(u) or simply by dG(u) or du. A large number of such indices depend only on vertex degree of the molecular graph.

One of the oldest and well known topological indices is the first and second Zagreb indices, was first introduced by Gutman et al. in 1972 [2], and it is defined as

M_{1} G = _{v∈V}d_{G}(v)^{2}= _{uv ∈E(G)} d_{G} u + d_{G}(v) (1)

M2 G = uv ∈E(G) dG u d_{G}(v) (2)
The first Zagreb polynomialM_{1} G, X and second Zagreb

polynomial M2 G, X are defined as:

M_{1} G, X = _{uv ∈E(G)}X^{ d}^{u}^{+d}^{v} (3)
M2 G, X = uv ∈E(G)X^{ d}^{u}^{×d}^{v} (4)
The properties of M_{1} G, X , M_{2} G, X polynomials for some

chemical structures have been studied in [3].The modified first and second Zagreb indices [4] are respectively defined a

M1

m G = _{d} ^{1}

G(u)^{2}

u∈V(G) (5) M2

m G = ^{1}

d_{G}(u)d_{G}(v)

uv ∈E(G) (6)

In [5], Shirdel et al. introduced the first hyper-Zagreb index of a graph G, which is defined as

HM1(G) = uv ∈E(G) d_{G} u + d_{G}(v) ^{2} (7)
In [6], the second hyper-Zagreb index of a graph G is defined

as

HM2(G) = uv ∈E(G) d_{G} u d_{G}(v) ^{2} (8)
In [7], M.Eliasi introduced the multiplicative first and second

Zagreb Indices defined as

π_{1}^{∗} G = _{uv ∈E(G)} d_{G} u + d_{G}(v) (9)
π2∗ G = uv ∈E(G)dG u d_{G}(v) (10)

The Geometric-arithmetic index, GA(G) index of a graph G
was introduced by D. Vukicevic et al. [8]. and it is defined as
GA = ^{2 d}_{d} ^{u}^{d}^{v}

ud_{v}

uv ∈E(G) (11)
The fifth Geometric-arithmetic index, GA_{5}(G) was introduced

by A.Graovac et al. [9] in 2011 which is defined as
GA5 G = ^{2 S}^{u}^{S}^{v}

S_{u}S_{v}

uv ∈E(G) (12)
Where Su is the sum of the degrees of all neighbors of the
vertex u in G, similarly for S_{ v}.

The atom-bond connectivity (ABC) index, proposed by Estrada et al. [10] and is defined as:

ABC G = ^{d}^{u}_{d}^{+d}^{v}^{−2}

ud_{v}

uv ∈E(G) (13) This index provides a good model for the stability of linear

and branched alkanes as well as the strain energy oF cycloalkanes [10,11]. Details about this index can be found in [12-14].

The fourth atom bond connectivity index, ABC4 G index was introduced by M. Ghorbani et al. in 2010.It is defined as

ABC4 G = ^{S}^{u}^{+S}^{v}^{−2}

S_{u}S_{v}

uv ∈E(G) (14) Where Su is the sum of the degrees of all neighbors of the

vertex u in G, similarly for S_{ v}. Further studies on ABC_{4} G
index can be found in [15].

Inspired by work on the ABC index, Furtula et al.

[16] proposed the following

**_______________________________________________________________________________________________**

**10 **
modified version of the ABC index and called it as augmented

**Zagreb index (AZI): **

AZI = _{d}^{d}^{u}^{d}^{v}

u+d_{v}−2
3

uv ∈E(G) (15) In [18], M.Eliasi introduced the multiplicative first and

second Zagreb Indices defined as

π1∗ G = G = uv ∈E(G) d_{G} u + d_{G}(v) (16)
π_{2}^{∗} G = G = _{uv ∈E(G)}d_{G} u d_{G}(v) (17)
In [17], V.R.Kulli introduced the multiplicative atom-bond

connectivity index and multiplicative Geometric-arithmetic indices which are defined as

ABCπ G = ^{d}^{u}_{d}^{+d}^{v}^{−2}

ud_{v}

uv ∈E(G) (18)
GAπ(G) = ^{2 d}_{d} ^{u}^{d}^{v}

u+d_{v}

uv ∈E(G) (19) Now we introduce ,

(i) The multiplicative first Zagreb polynomialM1π G, X and
second Zagreb polynomial M_{2}π G, X which is defined as

M1π G, X = uv ∈E(G)X^{ d}^{u}^{+d}^{v} (20)

M2π G, X = uv ∈E(G)X^{ d}^{u}^{×d}^{v} (21)
The multiplicative modified first and second Zagreb indices

defined as M1π

m G = ^{1}

d_{G} u ^{2}

u∈V G (22)
M_{2}π

m G = _{d} ^{1}

G(u)d_{G}(v)

uv ∈E(K_{n}) (23)

(iii) The multiplicative fifth Geometric-arithmetic index πGA5(G) is defined as

GA5π G = ^{2 S}_{S} ^{u}^{S}^{v}

u+S_{v}

uv ∈E(G) (24) (iv) The multiplicative fourth atom bond connectivity

index, ABC4 G index is defined as

ABC4π G = ^{S}^{u}_{S}^{+S}^{v}^{−2}

uS_{v}

uv ∈E(G) (25) (v) The multiplicative Augmented Zagreb index AZIπ(G) is

defined as

AZIπ(G) = _{d}^{d}^{u}^{d}^{v}

u+d_{v}−2
3

uv ∈E(G) (26)

**II. ** **C****OMPUTING MULTIPLICATIVE TOPOLOGICAL INDICES **
**OF SOME SPECIAL GRAPHS**

In this paper we calculate the first and second
Multiplicative Zagreb indices π1∗ G and π_{2}^{∗} G ,the first

and second multiplicative hyper Zagreb

indices HM1π G and HM2π(G) , first and second

Multiplicative modified Zagreb

indices Mm 1π G and M^{ m} _{2}π G , multiplicative atom-bond
connectivity index ABCπ G , multiplicative Geometric-
arithmetic index (GAπ G ) The multiplicative first Zagreb

polynomial and second Zagreb polynomial

((M1π G, X and M_{2}π G, X ), multiplicative ABC4π G index
, multiplicative GA5π G index and The multiplicative
Augmented Zagreb index (AZIπ(G)) of some special graphs .
**Lemma 2.1: **

Consider the complete graph Kn of order n, then

i π_{1}^{∗} K_{n} = 2 n − 1 ^{n (n −1)}^{2}

### π

_{2}

^{∗}

### K

_{n}

### = n − 1

^{n(n−1)}

### ii HM

_{1}π(K

_{n}) = 2 n − 1

^{n(n−1)}

HM2π(Kn) = n − 1 ^{2n(n−1)}

iii ^{m}M1π Kn = _{ n−1 }^{1} ^{n(n−1)}

^{m}M_{2}π K_{n} = _{ n−1 }^{1} ^{n(n−1)}

iv M_{1}π K_{n}, X = 2(n − 1) ^{n(n−1)}
M_{2}π K_{n}, X = X ^{n (n −1)3}^{2}

v AZIπ G = ^{(n−1)}_{2(n−2)}^{2}

3n (n −1)

2
vi ABCπ K_{n} = ^{1}_{2}

n

vii GAπ(K_{n}) = 1
viii ABC_{4}π Kn = ^{ 2n n−2 }_{ n−1 }_{2}

n (n −1)

2
xi GA_{5}π Kn = 1

**Proof. The degree of each vertex of a complete graph K**

_{n}of order n is n − 1 and the number of edges for Kn is equal to

1

2 n n − 1 or E(K_{n}) =^{1}_{2} n n − 1 . Then

i π1∗ Kn = uv ∈E(K_{n}) d_{G} u + dG(v)
= n − 1 + n − 1 ^{n(n−1)}^{2}

= 2 n − 1 ^{n(n−1)}^{2}
π2∗ K_{n} = uv ∈E(K_{n})dG u d_{G}(v)

= n − 1 × n − 1 ^{n(n−1)}^{2} = n − 1 ^{n(n−1)}

ii HM1π(Kn) = uv ∈E(K_{n}) d_{G} u + d_{G}(v) ^{2}
= n − 1 + n − 1 ^{2}

n(n−1) 2

= 2 n − 1 ^{n(n−1)}
HM2π Kn = uv ∈E(K_{n}) d_{G} u d_{G}(v) ^{2}

= n − 1 × n − 1 ^{2}

n (n −1) 2

= n − 1 ^{2n(n−1)}
iii M^{m} _{1}π K_{n} = _{d} ^{1}

G u ^{2}
u∈V G
= _{ n−1 }^{1} _{2}

n n −1

2 = _{ n−1 }^{1} ^{n(n−1)}
^{m}M_{2}π K_{n} = _{d} ^{1}

G(u)d_{G}(v)
uv ∈E(K_{n})
= n−1 × n−1 ^{1}

n n −1

2 = _{ n−1 }^{1} _{2}

n (n −1)

2 =

1 n−1

n(n−1)

(iv) M_{1}π G, X = M_{1}π K_{n}, X =
= uv ∈E(K_{n})X^{ d}^{u}^{+d}^{v}

**_______________________________________________________________________________________________**

**11 **
= X (n−1)+(n−1) ^{n (n −1)}_{2} = X^{2(n−1)} ^{n (n −1)}^{2} =

X ^{n(n−1)}^{2}

M_{2}π G, X = M_{2}π K_{n}, X
= _{uv ∈E(G)}X^{ d}^{u}^{×d}^{v}
= X (n−1)×(n−1)

uv ∈E(K_{n})

= X ^{n (n −1)3}^{2}
v AZIπ(K_{n}) = _{d}^{d}^{u}^{d}^{v}

u+d_{v}−2
3
uv ∈E(K_{n})

= (n−1)×(n−1) n−1 + n−1 −2

3 ^{n (n −1)}_{2}

=
= ^{(n−1)}_{2(n−2)}^{2}

3n (n −1)
2
vi ABCπ K_{n} = ^{d}^{u}_{d}^{+d}^{v}^{−2}

ud_{v}
uv ∈E(K_{n})
= n−1 + n−1 −2

(n−1)^{2}

n (n −1)

2 = ^{2(n−2)}_{(n−1)}_{2}

n (n −1)

2 =

2(n−2) (n−1)

n (n −1) 2

= n−1 + n−1 −2
(n−1)^{2}

n (n −1)

2 =

2(n−2)(n−1)2n(n−1)2=2(n−2)(n−1)n(n−1)2

vii GAπ(Kn) = ^{2 d}_{d} ^{u}^{d}^{v}

u+d_{v}
uv ∈E(K_{n})

= 2 n−1 × n−1 n−1 + n−1

n (n −1)

2 = 1

By the definition of ABC and GA, Su is the sum of the degrees
of all neighbors of the vertex u in GA and also S v, Hence we
have, S_{u}= S_{ v}= n − 1 × n − 1 = n − 1 ^{2}.Hence
viii ABC_{4}π Kn = ^{S}^{u}^{+S}^{v}^{−2}

S_{u}S_{v}
uv ∈E(K_{n})
= ^{ n−1 }_{ n−1 }^{2}^{+ n−1 }_{2}_{× n−1 }^{2}^{−2}_{2}

n (n −1)

2 =

^{𝟐 n−1 }_{ n−1 }^{2}_{4}^{−2}

n (n −1) 2

** **

### =

^{2 n−1 }

_{ n−1 }

^{2}

_{4}

^{−1 }

n (n −1)

2

### =

^{ 2 n}

_{ n−1 }

^{2}

^{−2n }

_{2}

n (n −1) 2

### =

^{ 2n n−2 }

_{ n−1 }

_{2}

n (n −1) 2

### xi GA5π Kn =

^{2 S}

^{u}

^{S}

^{v}

S_{u}+S_{v}
uv ∈E(K_{n}) ** **

### =

^{2 n−1 }

_{ n−1 }

_{2}

_{+ n−1 }

^{2}

^{× n−1 }

_{2}

^{2}

n (n −1)

2

### =

### =

^{2 n−1 }

_{2 n−1 }

_{2}

^{4}

n (n −1)

2

### =

^{2 n−1 }

_{2 n−1 }

_{2}

^{4}

n(n −1)

2

### = 1

**Lemma 2.2: **

Suppose C_{n} is a cycle of length n labeled by
1,2, … , n . Then

i π1∗ C_{n} = 4^{n}

### π

_{2}

^{∗}

### C

_{n}

### = 4

^{n}

ii HM1π(Cn) = 16^{n}

HM_{2}π(C_{n}) = 16^{n}

iii ^{m}M1π C_{n} = ^{1}_{4} ^{n}

^{m}M_{2}π C_{n} = ^{1}_{4} ^{n}
iv M_{1}π Cn, X =X^{4n}

M_{2}π C_{n}, X =X^{4n}
v AZIπ C_{n} =8^{n}
vi ABCπ C_{n} = ^{ 2(n−2)}_{(n−1)}

n (n −1)

2
vii GAπ(C_{n}) = 1
viii ABC_{4}π Cn = ^{𝟑}_{8}

n

xi GA5π Cn = 1
** **

**Proof. **

Here V(Cn) = n = E(Cn) and degree of each vertex is 2.Hence

i π1∗ C_{n} = uv ∈E(C_{n}) d_{G} u + d_{G}(v)
= 2 + 2 ^{n}= 4^{n}

π_{2}^{∗} C_{n} = _{uv ∈E(C}_{n}_{)}d_{G} u d_{G}(v)
= 2 × 2 ^{n} = 4^{n}

ii HM_{1}π(C_{n}= _{uv ∈E(C}_{n}_{)} d_{G} u + d_{G}(v) ^{2}
= 2 + 2 ^{2} ^{n}= 16^{n}

HM2π(Cn) = uv ∈E(C_{n}) d_{G} u dG(v) ^{2}
= 2 × 2 ^{2} ^{n} = 16^{n}
iii M^{m} 1π C_{n} = _{d} ^{1}

G u ^{2}
u∈V C_{n}
= _{2}^{1}_{2} ^{n} = ^{1}_{4}^{n}

mM2π C_{n} = ^{1}

d_{G}(u)d_{G}(v)
uv ∈E(C_{n})
= _{2×2}^{1} ^{n} = ^{1}_{4}^{n}
iv M1π G, X = M1π Cn, X =
= uv ∈E(C_{n})X^{ d}^{u}^{+d}^{v}

= X^{ 2+2 } ^{n}= X^{4n}

M_{2}π G, X = M_{2}π C_{n}, X = _{uv ∈E(C}_{n}_{)}X^{ d}^{u}^{×d}^{v}

**_______________________________________________________________________________________________**

**12 **
= uv ∈E(C_{n})X^{ 2×2 } = X^{4} ^{n}= X^{4n}

v AZIπ(C_{n}) = _{d}^{d}^{u}^{d}^{v}

u+d_{v}−2
3

uv ∈E(C_{n})

= _{2+2−2}^{2×2} ^{3} ^{n}= 8^{n}
vi ABCπ C_{n} = ^{d}^{u}^{+d}^{v}^{−2}

d_{u}d_{v}
uv ∈E(C_{n})

= ^{2+2−2}_{2×2}

n

= ^{1}_{2}

n

vii GAπ(Cn) = ^{2 d}_{d} ^{u}^{d}^{v}

u+d_{v}
uv ∈E(C_{n})
= ^{2 2×2}_{2+2} ^{n} = 1

By the definition of ABC and GA Su is the sum of the degrees
of all neighbors of the vertex u in G, similarly for S_{ v}, we have
Su= S v= 4.Hence

viii ABC_{4}π Cn = ^{S}^{u}_{S}^{+S}^{v}^{−2}

uS_{v}
uv ∈E(C_{n})

### =

^{4+4−2}

_{4×4}

n

### =

^{𝟑}

_{8}

n

** **

xi GA5π Cn = ^{2 S}_{S} ^{u}^{S}^{v}

u+S_{v}
uv ∈E(C_{n}) ** **

### =

^{2 4×4}

_{4+4}

^{n}

### = 1

**Lemma 2.3: **

Suppose Sn is the Star graph on n vertices,

i π1∗ S_{n} = n^{n−1}

### π

2∗### S

_{n}

### = n − 1

^{n−1}

### ii HM

_{1}π(S

_{n}) = n

^{2 n−1 }

HM_{2}π(S_{n}) = n − 1 ^{2 n−1 }

iii M^{m} _{1}π S_{n} =_{ n−1 }^{1} _{2}

^{m}M2π Sn = _{n−1}^{1} ^{n−1}

iv M1π Sn, X =X^{n n−1 }
M_{2}π S_{n}, X =X^{ n−1 }^{2}

v AZIπ S_{n} = ^{n−1}_{n−2} ^{3 n−1 }
vi ABCπ S_{n} = ^{n−2}

n−1 n−1

vii GAπ(S_{n}) = ^{2 n−1 }_{n} ^{n}

viii ABC_{4}π S_{n} = ^{ 2 n−2 }_{ n−1 }_{2} ^{n−1}
xi GA5π Sn = 1

**Proof : **

In a star Sn all the leaves are of degree 1 and the degree of internal node is

n − 1.Also the number of edges of S_{n} is n − 1 or E(S_{n}) =
n − 1 . Thus

i π_{1}^{∗}(S_{n})= _{uv ∈E(S}_{n}_{)} d_{G} u + d_{G}(v)
= 1 + n − 1 ^{n−1}= n^{n−1}

π2∗ S_{n} = uv ∈E(S_{n})dG u d_{G}(v)
= 1 × n − 1 ^{n−1}= n − 1 ^{n−1}
ii HM_{1}π(S_{n}) = _{uv ∈E(S}_{n}_{)} d_{G} u + d_{G}(v) ^{2}
= 1 + n − 1 ^{2} ^{n−1}= n^{2 n−1 }
HM_{2}π(S_{n}) = _{uv ∈E(S}_{n}_{)} d_{G} u d_{G}(v) ^{2}

= 1 × n − 1 ^{2} ^{n−1}

= n − 1 ^{2 n−1 }

iii M^{m} _{1}π Sn = ^{1}

d_{G} u ^{2}
u∈V S_{n}

= _{ n−1 }^{1} _{2} ^{1}× _{1}^{1}_{2} ^{n−1}=_{ n−1 }^{1} _{2}

mM2π Sn = ^{1}

d_{G}(u)d_{G}(v)
uv ∈E(S_{n})
= _{1× n−1 }^{1} ^{n−1}= _{n−1}^{1} ^{n−1}
iv M1π G, X = M_{1}π Sn, X

= uv ∈E(S_{n})X^{ d}^{u}^{+d}^{v}
= X^{ 1+ n−1 } ^{n−1}= X^{n n−1 }
M_{2}π G, X = M_{2}π S_{n}, X = _{uv ∈E(C}_{n}_{)}X^{ d}^{u}^{×d}^{v}

= X 1×n−1 = X^{n −1}^{n −1}=X^{ n−1 2}

uv ∈E(S_{n})

v AZIπ(S_{n}) = _{d}^{d}^{u}^{d}^{v}

u+d_{v}−2
3

uv ∈E(S_{n})

= _{uv ∈E(S}_{n}_{)} _{1+ n−1 −2}^{1×n−1} ^{3 n−1 }= ^{n−1}_{n−2} ^{3} ^{n−1}=
n−1n−23n−1

vi ABCπ S_{n} = ^{d}^{u}^{+d}^{v}^{−2}

d_{u}d_{v}
uv ∈E(S_{n})

= ^{1+ n−1 −2}_{1×n−1}

n−1

= ^{n−2}_{n−1}

n−1

vii GAπ(S_{n}) = ^{2 d}_{d} ^{u}^{d}^{v}

u+d_{v}
uv ∈E(S_{n})

= ^{2 1× n−1 }_{1+ n−1 }

n

= ^{2 n−1 }_{n}

n

By the definition of ABC and GA Su is the sum of the degrees of all neighbours of the vertex u in G, Let u be the central vertex and v be the non central vertex. Hence we have Su= S v= n − 1. Hence

viii ABC_{4}π S_{n} = ^{S}^{u}_{S}^{+S}^{v}^{−2}

uS_{v}
uv ∈E(S_{n}) ** **
= ^{n−1+n−1−2}_{n−1 × n−1}

n−1

**_______________________________________________________________________________________________**

**13 **

= _{ n−1 }^{2n−4}_{2}

n−1

= ^{ 2 n−2 }_{ n−1 }_{2} ^{n−1}

xi GA_{5}π Sn = ^{2 S}^{u}^{S}^{v}

S_{u}+S_{v}
uv ∈E(S_{n}) ** **
= 𝟐 n−1 × n−1

n−1 + n−1 n−1

** **
= ^{𝟐 n−1 }_{𝟐 n−1 }^{2} ^{n−1}**= 𝟏 **

**Lemma 2.4: **

If Wn, n ≥ 5 is the wheel graph on n vertices, then

i π1∗ W_{n} = n + 2 ^{ n−1 }× 6^{n−1}
π_{2}^{∗} W_{n} = 3 n − 1 ^{ n−1 }× 9^{ n−1 }

ii HM_{1}π(W_{n}) = n + 2 ^{2 n−1 }× 6^{2 n−1 }

HM_{2}π(W_{n}) = 3 n − 1 ^{2 n−1 }× 9^{2 n−1 }

iii M^{m} _{1} W_{n} = ^{1}_{9} ^{ n−1 }×_{ n−1 }^{1} _{2}
mM2 W_{n} = ^{1}

3n−3 n−1

× ^{1}_{9}^{ n−1 }
iv M1π Wn, X = X^{n}^{2}^{+7n−8}

M2π Wn, X =X^{3 n}^{2}^{−5n+2 }

v AZIπ Wn = ^{3 n−1 }_{n+1} ^{3 n−1 }× ^{9}_{4} ^{3 n−1 }

vi ABCπ W_{n} = _{3 n−1 }^{n}

n−1

× ^{2}_{3} ^{n−1}

vii GAπ(W_{n}) = ^{2 3 n−1 }_{n+2} ^{ n−1 }
viii ABC_{4}π Wn = _{3 n}_{2}^{ 4n }_{+4n−5 }

2 n−1

xi GA_{5}π Wn = ^{ 3 n}_{ 2n+1 }^{2}^{+4n−5 } ^{2n−1}

**Proof : The wheel graph 𝑊**𝑛 may denote the n-vertex graph
with an (n − 1)-cycle on its rim. For a wheel graph 𝑊_{𝑛}, 𝑛 ≥ 5 ,
it has 𝑉( 𝑊𝑛) = 𝑛 𝑎𝑛𝑑 𝐸( 𝑊𝑛) = 2 𝑛 − 1 .

Also the degree of end vertices of (n − 1) edges of a wheel graph 𝑊𝑛 is given by 3, (𝑛 − 1) and the remaining (n − 1) edges has the degree of end vertices as 3,3 . Then

i π_{1}^{∗} W_{n} = _{uv ∈E(W}_{n}_{)} d_{G} u + d_{G}(v)

### = 3 + n − 1

^{ n−1 }

### × 3 + 3

^{ n−1 }

### = n + 2

^{ n−1 }

### × 6

^{n−1}π2∗ W

_{n}= uv ∈E(W

_{n})dG u d

_{G}(v)

### = 3 × n − 1

^{ n−1 }

### × 3 × 3

^{ n−1 }

= 3 n − 1 ^{ n−1 }× 9^{ n−1 }
ii HM_{1}π(Wn) = uv ∈E(W_{n}) d_{G} u + d_{G}(v) ^{2}

### 3 + n − 1

^{2}

^{ n−1 }

### × 3 + 3

^{2}

^{ n−1 }

### = n + 2

^{2 n−1 }

### × 6

^{2 n−1 }HM2π(Wn) = uv ∈E(W

_{n}) d

_{G}u d

_{G}(v)

^{2}=

### 3 × n − 1

^{2}

^{ n−1 }

### × 3 × 3

^{2}

^{ n−1 }

### = 3 n − 1

^{2 n−1 }

### × 9

^{2 n−1 }iii M

^{m}

_{1}π W

_{n}=

^{1}

d_{G} u ^{2}
u∈V W_{n}

### =

_{ n−1 }

^{1}

_{2}

^{1}

### ×

_{3}

^{1}

_{2}

^{n−1}

### =

^{1}

_{9}

^{ n−1 }

### ×

_{ n−1 }

^{1}

_{2}

### M

^{m}2π W

_{n}=

^{1}

d_{G}(u)d_{G}(v)
uv ∈E(W_{n})

### =

_{3× n−1 }

^{1}

^{ n−1 }

### ×

_{3×3}

^{1}

^{ n−1 }

### =

_{3n−3}

^{1}

^{ n−1 }

### ×

^{1}

_{9}

^{ n−1 }

(iv) M_{1}π G, X = M_{1}π W_{n}, X = _{uv ∈E(W}_{n}_{)}X^{ d}^{u}^{+d}^{v}
= X^{ 3+(n−1) } ^{n−1}× X^{ 3+3 } ^{n−1}=
X^{ n+2 (n−1)}× X^{6(n−1)}

= X^{n}^{2}^{+n−2+6n−6}= X^{n}^{2}^{+7n−8}

M2π G, X = M_{2}π Wn, X = uv ∈E(C_{n})X^{ d}^{u}^{×d}^{v}
= X^{ 3×(n−1) } ^{n−1}× X^{ 3×3 } ^{n−1}=
X 3n−3 (n−1)× X^{9(n−1)}

= X^{3n}^{2}^{−6n+3−9n−9}= X^{3n}^{2}^{−15n+6}

= X^{3 n}^{2}^{−5n+2 }
v AZIπ(Wn) = _{d}^{d}^{u}^{d}^{v}

u+d_{v}−2
3

uv ∈E(W_{n})

= _{3+ n−1 −2}^{3×n−1} ^{3 n−1 }× _{3+3−2}^{3×3} ^{3 n−1 }=
^{3 n−1 }_{n+1} ^{3 n−1 }× ^{9}_{4} ^{3 n−1 }

vi ABCπ W_{n} = ^{d}^{u}_{d}^{+d}^{v}^{−2}

ud_{v}
uv ∈E(W_{n})

= ^{3+(n−1)−2}_{3×(n−1)}

n−1

× ^{3+3−2}_{3×3}

n−1

= n3n−1n−1×23n−1

vii GAπ(W_{n}) = ^{2 d}_{d} ^{u}^{d}^{v}

u+d_{v}
uv ∈E(W_{n})

= ^{2 3× n−1 }_{3+ n−1 } ^{ n−1 }× ^{2 3×3}_{3+3} ^{ n−1 }=

2 3 n−1 n+2

n−1

By the definition of ABC and GA Su is the sum of the degrees
of all neighbours of the vertex u in G, similarly for S v, we
have S_{u}= n − 1 ; S_{ v}= 3 n − 1 .Hence

viii ABC_{4}π Wn = ^{S}^{u}^{+S}^{v}^{−2}

S_{u}S_{v}
uv ∈E(W_{n})

**_______________________________________________________________________________________________**

**14 **

### =

n+5 +3 n−1 −2 n+5 ×3 n−12 n−1

### =

_{3 n}

_{2}

^{ 4n }

_{+4n−5 }

2 n−1

### xi GA

_{5}π W

_{n}=

^{2 S}

_{S}

^{u}

^{S}

^{v}

u+S_{v}
uv ∈E(W_{n})

### =

2 n+5 ×3 n−1n+5 +3 n−1 2n−1

### =

2 3 n^{2}+4n−5
4n+2

2n−1

### =

^{ 3 n}

_{ 2n+1 }

^{2}

^{+4n−5 }

^{2n−1}

**Lemma 2.5: **

If F_{n}, n ≥ 2 is the fan graph on n vertices, then

i π_{1}^{∗} F_{n} = 25 × 6^{n−3}× 2 + n ^{2}× 3 + n ^{n−2}
π2∗ F_{n} = 36 × 9^{n−3}× 4n^{2}× 3n ^{n−2}
ii HM1π(Fn) = 625 × 2 + n ^{4}× 3 + n ^{2n−4}× 36^{n−3}
HM2π(Fn) = 1296 × 16n^{4}× 3n ^{2n−4}× 81^{n−3}
iii M^{m} 1π Fn = _{16}^{1} _{n}^{1}_{2} ^{1}_{9} ^{n−2}

M^{m} _{2}π F_{n} = _{36}^{1} × _{4n}^{1}_{2} × _{3n}^{1} ^{n−2}× ^{1}_{9} ^{n−3}
iv M_{1}π F_{n}, X = 𝑋^{2𝑛}^{2}^{+11𝑛+5}

M2π Fn, X =𝑋^{7𝑛}^{2}^{+3𝑛+9}
v AZIπ F_{n} =4096 _{n+1}^{3n} ^{3n−6}× ^{9}_{4} ^{3n−9}
vi ABCπ F_{n} =^{1}

4 ^{n+1}_{3n}

n−2

× ^{2}_{3} ^{n−3}

vii GAπ(K_{n}) =^{24}_{25} _{ n+2 }^{8n} _{2} ^{2 3n}_{n+3} ^{n−2}

viii ABC_{4}π Fn = ^{1}

36 _{3n}^{4}

n 2 3

n−4

xi GA_{5}π Fn =

^{3n n−2 }_{2 n−3 }

n

× ^{2 n+3 }_{ n+5 }

4

× ^{3 n+5 }_{ n+8 }

4

×

^{3}_{2} ^{n−4}

**Proof : For a fan graph** 𝐹_{𝑛}, 𝑛 ≥ 2 , 𝑉( 𝐹_{𝑛}) = 𝑛 +
1 𝑎𝑛𝑑 𝐸( 𝐹𝑛) = 2𝑛 − 1. Also in 𝐹𝑛 the degrees of end
vertices of 𝑛 − 2 edges are 3, 𝑛 , 2 edges have the degrees
of the end vertices as 3,2 , 2 edges have the degrees of the
end vertices as 𝑛, 2 and the remaining 𝑛 − 3 edges have the
degrees of the end vertices as 3,3 . Then

### i π

_{1}

^{∗}

### F

_{n}

### =

_{uv ∈E(F}

_{n}

_{)}

### d

_{G}

### u + d

_{G}

### (v)

### = 2 + 3

^{2}

### × 2 + n

^{2}

### × 3 + n

^{n−2}

### × 3 + 3

^{n−3}

### = 25 × 6

^{n−3}

### × 2 + n

^{2}

### × 3 + n

^{n−2}π

_{2}

^{∗}F

_{n}=

_{uv ∈E(F}

_{n}

_{)}d

_{G}u d

_{G}(v)

### = 2 × 3

^{2}

### × 2 × n

^{2}

### × 3 × n

^{n−2}

### × 3 × 3

^{n−3}

### = 36 × 9

^{n−3}

### × 4n

^{2}

### × 3n

^{n−2}ii HM

_{1}π(Fn) = uv ∈E(F

_{n}) d

_{G}u + d

_{G}(v)

^{2}

### = 2 + 3

^{2}

^{2}

### × 2 + n

^{2}

^{2}

### × 3 + n

^{2}

^{n−2}

### × 3 + 3

^{2}

^{n−3}

### = 625 × 2 + n

^{4}

### × 3 + n

^{2n−4}

### × 36

^{n−3}HM2π(Fn) = uv ∈E(F

_{n}) d

_{G}u d

_{G}(v)

^{2}

### = 2 × 3

^{2}

^{2}

### × 2 × n

^{2}

^{2}

### × 3 × n

^{2}

^{n−2}

### × 3 × 3

^{2}

^{n−3}

### 1296 × 16n

^{4}

### × 3n

^{2n−4}

### × 81

^{n−3}iii M

^{m}

_{1}π F

_{n}=

^{1}

d_{G} u ^{2}
u∈V F_{n}

= _{n}^{1}_{2} ^{1}× _{2}^{1}_{2} ^{2}× _{3}^{1}_{2} ^{n−2}=_{16}^{1} _{n}^{1}_{2} ^{1}_{9} ^{n−2}
M^{m} 2π F_{n} = _{d} ^{1}

G(u)d_{G}(v)
uv ∈E(F_{n})

### =

_{2×3}

^{1}

^{2}

### ×

_{2×n}

^{1}

^{2}

### ×

_{3×n}

^{1}

^{n−2}

### ×

_{3×3}

^{1}

^{n−3}

### =

_{36}

^{1}

### ×

_{4n}

^{1}

_{2}

### ×

_{3n}

^{1}

^{n−2}

### ×

^{1}

_{9}

^{n−3}

### (iv) M

_{1}π G, X = M

_{1}π F

_{n}, X =

uv ∈E(W_{n})X^{ d}^{u}^{+d}^{v}

= X^{ n+2 }^{2}× X^{ 3+2 } ^{2}× X^{ n+3 }^{n−2}× X^{ 3+3 } ^{n−3}
= 𝑋^{ 𝑛}^{2}^{+4𝑛+4+25+𝑛}^{2}+𝑛−6+6𝑛−18 = 𝑋^{2𝑛}^{2}^{+11𝑛+5}

M2π G, X = M2π Fn, X =
uv ∈E(C_{n})X^{ d}^{u}^{×d}^{v}
= X^{ n×2 }^{2}× X^{ 3×2 } ^{2}× X^{ n×3 } ^{n−2}× X^{ 3×3 } ^{n−3}
= 𝑋^{ 4𝑛}^{2}^{+36+3𝑛}^{2}^{−6𝑛+9𝑛−27 }= 𝑋^{7𝑛}^{2}^{+3𝑛+9}
v AZIπ(F_{n}) = _{d}^{d}^{u}^{d}^{v}

u+d_{v}−2
3

uv ∈E(F_{n})

### =

_{2+3−2}

^{2×3}

^{3}

^{2}

### ×

_{n+2−2}

^{n×2}

^{3}

2

### ×

_{n+3−2}

^{n×3}

^{3}

^{n−2}

### ×

_{3+3−2}

^{3×3}

^{3}

^{n−3}

### = 2

^{6}

### × 2

^{6}

### ×

_{n+1}

^{3n}

^{3n−6}

### ×

_{4}

^{9}

^{3n−9}

### = 4096

^{3n}

n+1

3n−6

### ×

^{9}

4 3n−9

### vi ABCπ F

_{n}=

^{d}

^{u}

^{+d}

^{v}

^{−2}

d_{u}d_{v}
uv ∈E(F_{n})
= ^{2+n−2}_{2×n}

2

× ^{2+3−2}_{2×3}

2

× ^{3+n−2}_{3×n}

n−2

× ^{3+3−2}_{3×3}

n−3

=_{ 4}^{1}× ^{n+1}_{3n}

n−2

× ^{2}_{3} ^{n−3}

vii GAπ(F_{n}) = ^{2 d}_{d} ^{u}^{d}^{v}

u+d_{v}
uv ∈E(F_{n})