# On Some Multiplicative Topological Indices

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## 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

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I. INTRODUCTION

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

M1 G = v∈VdG(v)2= uv ∈E(G) dG u + dG(v) (1)

M2 G = uv ∈E(G) dG u dG(v) (2) The first Zagreb polynomialM1 G, X and second Zagreb

polynomial M2 G, X are defined as:

M1 G, X = uv ∈E(G)X du+dv (3) M2 G, X = uv ∈E(G)X du×dv (4) The properties of M1 G, X , M2 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

dG(u)dG(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) dG u + dG(v) 2 (7) In [6], the second hyper-Zagreb index of a graph G is defined

as

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

Zagreb Indices defined as

π1 G = uv ∈E(G) dG u + dG(v) (9) π2 G = uv ∈E(G)dG u dG(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 dd udv

udv

uv ∈E(G) (11) The fifth Geometric-arithmetic index, GA5(G) was introduced

by A.Graovac et al. [9] in 2011 which is defined as GA5 G = 2 SuSv

SuSv

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 = dud+dv−2

udv

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 = Su+Sv−2

SuSv

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 ABC4 G index can be found in [15].

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

[16] proposed the following

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

Zagreb index (AZI):

AZI = ddudv

u+dv−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) dG u + dG(v) (16) π2 G = G = uv ∈E(G)dG u dG(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 = dud+dv−2

udv

uv ∈E(G) (18) GAπ(G) = 2 dd udv

u+dv

uv ∈E(G) (19) Now we introduce ,

(i) The multiplicative first Zagreb polynomialM1π G, X and second Zagreb polynomial M2π G, X which is defined as

M1π G, X = uv ∈E(G)X du+dv (20)

M2π G, X = uv ∈E(G)X du×dv (21) The multiplicative modified first and second Zagreb indices

defined as M1π

m G = 1

dG u 2

u∈V G (22) M2π

m G = d 1

G(u)dG(v)

uv ∈E(Kn) (23)

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

GA5π G = 2 SS uSv

u+Sv

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

index, ABC4 G index is defined as

ABC4π G = SuS+Sv−2

uSv

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

defined as

AZIπ(G) = ddudv

u+dv−2 3

uv ∈E(G) (26)

II. COMPUTING MULTIPLICATIVE TOPOLOGICAL INDICES OF SOME SPECIAL GRAPHS

In this paper we calculate the first and second Multiplicative Zagreb indices π1 G and π2 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 2π 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 M2π 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 π1 Kn = 2 n − 1 n (n −1)2

2

n

### = n − 1

n(n−1) ii HM1π(Kn) = 2 n − 1 n(n−1)

HM2π(Kn) = n − 1 2n(n−1)

iii mM1π Kn = n−1 1 n(n−1)

mM2π Kn = n−1 1 n(n−1)

iv M1π Kn, X = 2(n − 1) n(n−1) M2π Kn, X = X n (n −1)32

v AZIπ G = (n−1)2(n−2)2

3n (n −1)

2 vi ABCπ Kn = 12

n

vii GAπ(Kn) = 1 viii ABC4π Kn = 2n n−2 n−1 2

n (n −1)

2 xi GA5π Kn = 1

Proof. The degree of each vertex of a complete graph Kn of order n is n − 1 and the number of edges for Kn is equal to

1

2 n n − 1 or E(Kn) =12 n n − 1 . Then

i π1 Kn = uv ∈E(Kn) dG u + dG(v) = n − 1 + n − 1 n(n−1)2

= 2 n − 1 n(n−1)2 π2 Kn = uv ∈E(Kn)dG u dG(v)

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

ii HM1π(Kn) = uv ∈E(Kn) dG u + dG(v) 2 = n − 1 + n − 1 2

n(n−1) 2

= 2 n − 1 n(n−1) HM2π Kn = uv ∈E(Kn) dG u dG(v) 2

= n − 1 × n − 1 2

n (n −1) 2

= n − 1 2n(n−1) iii Mm 1π Kn = d 1

G u 2 u∈V G = n−1 1 2

n n −1

2 = n−1 1 n(n−1) mM2π Kn = d 1

G(u)dG(v) uv ∈E(Kn) = n−1 × n−1 1

n n −1

2 = n−1 1 2

n (n −1)

2 =

1 n−1

n(n−1)

(iv) M1π G, X = M1π Kn, X = = uv ∈E(Kn)X du+dv

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11 = X (n−1)+(n−1) n (n −1)2 = X2(n−1) n (n −1)2 =

X n(n−1)2

M2π G, X = M2π Kn, X = uv ∈E(G)X du×dv = X (n−1)×(n−1)

uv ∈E(Kn)

= X n (n −1)32 v AZIπ(Kn) = ddudv

u+dv−2 3 uv ∈E(Kn)

= (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 Kn = dud+dv−2

udv uv ∈E(Kn) = 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 dd udv

u+dv uv ∈E(Kn)

= 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, Su= S v= n − 1 × n − 1 = n − 1 2.Hence viii ABC4π Kn = Su+Sv−2

SuSv uv ∈E(Kn) = n−1 n−1 2+ n−1 2× n−1 2−22

n (n −1)

2 =

𝟐 n−1 n−1 24−2

n (n −1) 2

2 n−1 n−1 24−1

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 SuSv

Su+Sv uv ∈E(Kn)

### =

2 n−1 n−1 2+ n−1 2× n−1 22

n (n −1)

2

2 n−1 2 n−1 24

n (n −1)

2

2 n−1 2 n−1 24

n(n −1)

2

### = 1

Lemma 2.2:

Suppose Cn is a cycle of length n labeled by 1,2, … , n . Then

i π1 Cn = 4n

2

n

### = 4

n

ii HM1π(Cn) = 16n

HM2π(Cn) = 16n

iii mM1π Cn = 14 n

mM2π Cn = 14 n iv M1π Cn, X =X4n

M2π Cn, X =X4n v AZIπ Cn =8n vi ABCπ Cn = 2(n−2)(n−1)

n (n −1)

2 vii GAπ(Cn) = 1 viii ABC4π Cn = 𝟑8

n

xi GA5π Cn = 1

Proof.

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

i π1 Cn = uv ∈E(Cn) dG u + dG(v) = 2 + 2 n= 4n

π2 Cn = uv ∈E(Cn)dG u dG(v) = 2 × 2 n = 4n

ii HM1π(Cn= uv ∈E(Cn) dG u + dG(v) 2 = 2 + 2 2 n= 16n

HM2π(Cn) = uv ∈E(Cn) dG u dG(v) 2 = 2 × 2 2 n = 16n iii Mm 1π Cn = d 1

G u 2 u∈V Cn = 212 n = 14n

mM2π Cn = 1

dG(u)dG(v) uv ∈E(Cn) = 2×21 n = 14n iv M1π G, X = M1π Cn, X = = uv ∈E(Cn)X du+dv

= X 2+2 n= X4n

M2π G, X = M2π Cn, X = uv ∈E(Cn)X du×dv

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### _______________________________________________________________________________________________

12 = uv ∈E(Cn)X 2×2 = X4 n= X4n

v AZIπ(Cn) = ddudv

u+dv−2 3

uv ∈E(Cn)

= 2+2−22×2 3 n= 8n vi ABCπ Cn = du+dv−2

dudv uv ∈E(Cn)

= 2+2−22×2

n

= 12

n

vii GAπ(Cn) = 2 dd udv

u+dv uv ∈E(Cn) = 2 2×22+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 ABC4π Cn = SuS+Sv−2

uSv uv ∈E(Cn)

4+4−24×4

n

### =

𝟑8

n

xi GA5π Cn = 2 SS uSv

u+Sv uv ∈E(Cn)

2 4×44+4 n

### = 1

Lemma 2.3:

Suppose Sn is the Star graph on n vertices,

i π1 Sn = nn−1

2

n

### = n − 1

n−1 ii HM1π(Sn) = n2 n−1

HM2π(Sn) = n − 1 2 n−1

iii Mm 1π Sn = n−1 1 2

mM2π Sn = n−11 n−1

iv M1π Sn, X =Xn n−1 M2π Sn, X =X n−1 2

v AZIπ Sn = n−1n−2 3 n−1 vi ABCπ Sn = n−2

n−1 n−1

vii GAπ(Sn) = 2 n−1 n n

viii ABC4π Sn = 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 Sn is n − 1 or E(Sn) = n − 1 . Thus

i π1(Sn)= uv ∈E(Sn) dG u + dG(v) = 1 + n − 1 n−1= nn−1

π2 Sn = uv ∈E(Sn)dG u dG(v) = 1 × n − 1 n−1= n − 1 n−1 ii HM1π(Sn) = uv ∈E(Sn) dG u + dG(v) 2 = 1 + n − 1 2 n−1= n2 n−1 HM2π(Sn) = uv ∈E(Sn) dG u dG(v) 2

= 1 × n − 1 2 n−1

= n − 1 2 n−1

iii Mm 1π Sn = 1

dG u 2 u∈V Sn

= n−1 1 2 1× 112 n−1= n−1 1 2

mM2π Sn = 1

dG(u)dG(v) uv ∈E(Sn) = 1× n−1 1 n−1= n−11 n−1 iv M1π G, X = M1π Sn, X

= uv ∈E(Sn)X du+dv = X 1+ n−1 n−1= Xn n−1 M2π G, X = M2π Sn, X = uv ∈E(Cn)X du×dv

= X 1×n−1 = Xn −1n −1=X n−1 2

uv ∈E(Sn)

v AZIπ(Sn) = ddudv

u+dv−2 3

uv ∈E(Sn)

= uv ∈E(Sn) 1+ n−1 −21×n−1 3 n−1 = n−1n−2 3 n−1= n−1n−23n−1

vi ABCπ Sn = du+dv−2

dudv uv ∈E(Sn)

= 1+ n−1 −21×n−1

n−1

= n−2n−1

n−1

vii GAπ(Sn) = 2 dd udv

u+dv uv ∈E(Sn)

= 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 ABC4π Sn = SuS+Sv−2

uSv uv ∈E(Sn) = n−1+n−1−2n−1 × n−1

n−1

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13

= n−1 2n−42

n−1

= 2 n−2 n−1 2 n−1

xi GA5π Sn = 2 SuSv

Su+Sv uv ∈E(Sn) = 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 Wn = n + 2 n−1 × 6n−1 π2 Wn = 3 n − 1 n−1 × 9 n−1

ii HM1π(Wn) = n + 2 2 n−1 × 62 n−1

HM2π(Wn) = 3 n − 1 2 n−1 × 92 n−1

iii Mm 1 Wn = 19 n−1 × n−1 1 2 mM2 Wn = 1

3n−3 n−1

× 19 n−1 iv M1π Wn, X = Xn2+7n−8

M2π Wn, X =X3 n2−5n+2

v AZIπ Wn = 3 n−1 n+1 3 n−1 × 94 3 n−1

vi ABCπ Wn = 3 n−1 n

n−1

× 23 n−1

vii GAπ(Wn) = 2 3 n−1 n+2 n−1 viii ABC4π Wn = 3 n2 4n +4n−5

2 n−1

xi GA5π Wn = 3 n 2n+1 2+4n−5 2n−1

Proof : The wheel graph 𝑊𝑛 may denote the n-vertex graph with an (n − 1)-cycle on its rim. For a wheel graph 𝑊𝑛, 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 π1 Wn = uv ∈E(Wn) dG u + dG(v)

n−1

n−1

n−1

### × 6

n−1 π2 Wn = uv ∈E(Wn)dG u dG(v)

n−1

### × 3 × 3

n−1

= 3 n − 1 n−1 × 9 n−1 ii HM1π(Wn) = uv ∈E(Wn) dG u + dG(v) 2

2 n−1

2 n−1

2 n−1

### × 6

2 n−1 HM2π(Wn) = uv ∈E(Wn) dG u dG(v) 2=

2 n−1

2 n−1

2 n−1

### × 9

2 n−1 iii Mm 1π Wn = 1

dG u 2 u∈V Wn

n−1 1 2 1

312 n−1

19 n−1

### ×

n−1 1 2 Mm 2π Wn = 1

dG(u)dG(v) uv ∈E(Wn)

3× n−1 1 n−1

3×31 n−1

3n−31 n−1

### ×

19 n−1

(iv) M1π G, X = M1π Wn, X = uv ∈E(Wn)X du+dv = X 3+(n−1) n−1× X 3+3 n−1= X n+2 (n−1)× X6(n−1)

= Xn2+n−2+6n−6= Xn2+7n−8

M2π G, X = M2π Wn, X = uv ∈E(Cn)X du×dv = X 3×(n−1) n−1× X 3×3 n−1= X 3n−3 (n−1)× X9(n−1)

= X3n2−6n+3−9n−9= X3n2−15n+6

= X3 n2−5n+2 v AZIπ(Wn) = ddudv

u+dv−2 3

uv ∈E(Wn)

= 3+ n−1 −23×n−1 3 n−1 × 3+3−23×3 3 n−1 = 3 n−1 n+1 3 n−1 × 94 3 n−1

vi ABCπ Wn = dud+dv−2

udv uv ∈E(Wn)

= 3+(n−1)−23×(n−1)

n−1

× 3+3−23×3

n−1

= n3n−1n−1×23n−1

vii GAπ(Wn) = 2 dd udv

u+dv uv ∈E(Wn)

= 2 3× n−1 3+ n−1 n−1 × 2 3×33+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 Su= n − 1 ; S v= 3 n − 1 .Hence

viii ABC4π Wn = Su+Sv−2

SuSv uv ∈E(Wn)

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14

### =

n+5 +3 n−1 −2 n+5 ×3 n−1

2 n−1

### =

3 n2 4n +4n−5

2 n−1

xi GA5π Wn = 2 SS uSv

u+Sv uv ∈E(Wn)

2 n+5 ×3 n−1

n+5 +3 n−1 2n−1

2 3 n2+4n−5 4n+2

2n−1

### =

3 n 2n+1 2+4n−5 2n−1

Lemma 2.5:

If Fn, n ≥ 2 is the fan graph on n vertices, then

i π1 Fn = 25 × 6n−3× 2 + n 2× 3 + n n−2 π2 Fn = 36 × 9n−3× 4n2× 3n n−2 ii HM1π(Fn) = 625 × 2 + n 4× 3 + n 2n−4× 36n−3 HM2π(Fn) = 1296 × 16n4× 3n 2n−4× 81n−3 iii Mm 1π Fn = 161 n12 19 n−2

Mm 2π Fn = 361 × 4n12 × 3n1 n−2× 19 n−3 iv M1π Fn, X = 𝑋2𝑛2+11𝑛+5

M2π Fn, X =𝑋7𝑛2+3𝑛+9 v AZIπ Fn =4096 n+13n 3n−6× 94 3n−9 vi ABCπ Fn =1

4 n+13n

n−2

× 23 n−3

vii GAπ(Kn) =2425 n+2 8n 2 2 3nn+3 n−2

viii ABC4π Fn = 1

36 3n4

n 2 3

n−4

xi GA5π Fn =

3n n−2 2 n−3

n

× 2 n+3 n+5

4

× 3 n+5 n+8

4

×

32 n−4

Proof : For a fan graph 𝐹𝑛, 𝑛 ≥ 2 , 𝑉( 𝐹𝑛) = 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

1

n

uv ∈E(Fn)

G

G

2

2

n−2

n−3

n−3

2

### × 3 + n

n−2 π2 Fn = uv ∈E(Fn)dG u dG(v)

2

2

n−2

n−3

n−3

2

### × 3n

n−2 ii HM1π(Fn) = uv ∈E(Fn) dG u + dG(v) 2

2 2

2 2

2 n−2

2 n−3

4

2n−4

### × 36

n−3 HM2π(Fn) = uv ∈E(Fn) dG u dG(v) 2

2 2

2 2

2 n−2

2 n−3

4

2n−4

### × 81

n−3 iii Mm 1π Fn = 1

dG u 2 u∈V Fn

= n12 1× 212 2× 312 n−2=161 n12 19 n−2 Mm 2π Fn = d 1

G(u)dG(v) uv ∈E(Fn)

2×31 2

2×n1 2

3×n1 n−2

3×31 n−3

361

4n12

3n1 n−2

### ×

19 n−3 (iv) M1π G, X = M1π Fn, X =

uv ∈E(Wn)X du+dv

= X n+2 2× X 3+2 2× X n+3 n−2× X 3+3 n−3 = 𝑋 𝑛2+4𝑛+4+25+𝑛2+𝑛−6+6𝑛−18 = 𝑋2𝑛2+11𝑛+5

M2π G, X = M2π Fn, X = uv ∈E(Cn)X du×dv = X n×2 2× X 3×2 2× X n×3 n−2× X 3×3 n−3 = 𝑋 4𝑛2+36+3𝑛2−6𝑛+9𝑛−27 = 𝑋7𝑛2+3𝑛+9 v AZIπ(Fn) = ddudv

u+dv−2 3

uv ∈E(Fn)

2+3−22×3 3 2

n+2−2n×2 3

2

n+3−2n×3 3 n−2

3+3−23×3 3 n−3

6

6

n+13n 3n−6

49 3n−9

3n

n+1

3n−6

### ×

9

4 3n−9

vi ABCπ Fn = du+dv−2

dudv uv ∈E(Fn) = 2+n−22×n

2

× 2+3−22×3

2

× 3+n−23×n

n−2

× 3+3−23×3

n−3

= 41× n+13n

n−2

× 23 n−3

vii GAπ(Fn) = 2 dd udv

u+dv uv ∈E(Fn)

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## References

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