Vol 7, No 12 (2017)

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Research Article

a

December

2017

Computer Science and Software Engineering

ISSN: 2277-128X (Volume-7, Issue-12)

On Topological Indices of Sudoku Graphs and Titania

TiO

₂

Nanotubes

K. Pattabiraman*

Department of Mathematics, Annamalai University, Annamalainagar, Tamilnadu, India

A. Santhakumar

Department of Mathematics, CK College of Engineering and Technology, Cuddalore, Tamilnadu, India

Abstract— In this paper, we obtain the exact formulae for some topological indices such as the general sum-connectivity index, atom-bond sum-connectivity index, geometric arithmetic index, inverse sum indeg index, symmetric division deg index and harmonic polynomial of titania TiO2 Nanotubes and Sudoku graphs.

Keywords— sum-connectivity index, Sudoku graph, TiO2 Nanotubes. MSC: Primary: 05C35; Secondary: 05C07, 05C40

I. INTRODUCTION

A topological index is a mathematical measure which correlates to the chemical structures of any simple finite graph. They are invariant under the graph isomorphism. They play an important role in the study of QS AR / QS PR. In theoretical chemistry, molecular structure descriptors (also called topological indices) are used for modeling physicochemical, pharmacologic, toxicologic, nanoscience, biological and other properties of chemical compounds. Among these topological descriptors the degree-based topological indices are of great importance. The first degree-based topological indices that were defined by Gutman and Trinajstić in [8] 1972, are the first and second Zagreb indices. These indices were originally defined as M1(G) =





2





( ) ( )

( ) ( ) ( )

G G G

u V G uv E G

d u d u d v

 

 



and

M2(G) =

( )

( ) ( )

G G uv E G

d u d v





, where M1 (G) and M2(G) denote the first and second Zagreb indices, respectively.

The sum-connectivity index was proposed by Zhou and Trinajstić [15] in 2009, which is defined as the sum over

all the edges of the graph of the terms

1 2

(

d u

_G

( )

d

_G

( )) .

v





This concept was extended to the general sum-connectivity

index in 2010 [16], which is defined as

( )

( ) ( _G( ) _G( )) ,

uv E G

G d u d v 











 where α is a real number. If α = 1, 1 2



and 2, then



₁(G) =

M

₁ (G), ₁ 2

( )

G

( )

G



_





and



₂(G) = HM (G) (hyper Zagreb index). The sum-connectivity

index and product-connectivity index correlate well with the π- electron energy of benzenoid hydrocarbons [12]. Another

variant of the Randic index of G is the harmonic index, denoted by H(G) and define as

H(G) =

1 2 ( ) 1 2

2

2 1( ). ( ) ( )

a a E G G G

G

d a d a 



 





Motivated by definition of the Randić connectivity index, Vukicević and Furtula [14] proposed another vertedx-degree-based topological index, named the geometric-arithmetic index. The geometric-arithmetic index of a graph G is

denoted by GA(G) and defined as GA(G) =

( )

2 ( ) ( )

.

( ) ( )

G G uv E G G G

d u d v

d u d v







One of the well-know degree based topological index is the atom-bond connectivity (ABC) index of G, proposed

by Estrada et al. in [4], and defined as ABC(G) =

( )

( ) ( ) 2 . ( ) ( )

G G

uv E G G G

d u d v d u d v



 



The inverse sum indeg index IS I (G) of a simple graph G is defined [5] in as IS I (G) =

( )

( ) ( ) . ( ) ( )

G G uv E G G G

d u d v d u d v







The symmetric division deg index of a connected graph G, (S DD) is defined in [7] as S

DD (G) = 2 2

( )

( ) ( ) . ( ) ( )

G G

uv E G G G

d u d v d u d v







The harmonic polynomial is defined in [9] as H (G, x) = ( ) ( ) 1

( )

2

dG u dG v

.

uv E G

x

 





Note that 1

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ISSN(E): 2277-128X, ISSN(P): 2277-6451, pp. 96-103

In this paper, we obtain the exact formulae for some topological indices such as the general sum-connectivity index, atom-bond connectivity index, geometric arithmetic index, inverse sum indeg index, symmetric division deg index and harmonic polynomial of titania TiO2 nanotubes and Sudoku graphs.

II. n x n SUDOKU GRAPHS

Sudoku is a popular puzzle game. An n x n sudoku puzzle is a grid of cells partitioned into n smaller block of n

element. Sudoku can also be viewed as a bipartite graph. From the structure of n x n sodoku graph, the edge set can be partitioned into eight sets show in Table 1

Theoren 2.1.Consider G = (SK) n x nis a sudoku graph for n ≥ 2, Then its general sum connectivity index is



 (G) =

(12)α 8 + (13)α 12+(13)α 8n + (12)α (4n – 4) + (14)α (32n – 40) + (14)α 4n2 + (15)α (20n2 – 36n + 20) + (16)α (10n2 – 14n + 4).

Proof:Now we compute the sum connectivity index of n x n sudoku graph and by the definition of



_, we have

, ( )

(

)

(

_G

( )

_G

( ))

v u E G

G

d

u

d

v













= (12)α |E1| + (13)α |E2| + (12)α |E3| + (13)α |E4| + (14)α |E5|+ (14)α |E6| + (15)α |E7| + (16)α |E8| Hence



(G)= (12)α 8 + (13)α 12 + (13)α 8n + (12)α (4n – 4) + (14)α (32n – 40)

+ (14)α 4n2 + (15)α (20n2 – 36n + 20) + (16)α (10n2 – 14n + 4).■

Table I

(dG(u), dG(v)) where uv



E(G) Number of edges (5, 7)

(5, 8) (6, 6) (6, 7) (6, 8) (7, 7) (7, 8) (8, 8)

8 12 4n-4

8n

32n – 40 4n2

20n2 – 36n + 20 10n2 – 14n + 4 Table 1. Edge partition of sudoku (S K)1 X 1 based on degrees of end vertices of each edge.

Theorem 2.2. Using the above theorem,we obtain the following result Let G = (S K)n x nsudoku graph for n ≥ 2, Then

4(129n2 – 41n + 2), if α = 1; 2

3055 362 6380 , 1560

n  n _if _{α =}_-1;





(G) = 1385n2 – 87n + 72, if α = 2;



2



₂

20 36 20

2 2 12 8 5 7 2

,

3 13 15 2

n n

n _  n_   _ n  n _if _{α =} 1_;

2



Proof:Putting α = 1 in Theorem 2.1, we get



(G) = (12)8 + (13)12 + (13)8n + (12) (4n – 4) + (14) (32n – 40) + (14)4n2 +(15) (20n2 – 36n+ 20) + (16) (10n2 – 14n+4)

= 4(129n2 – 41n + 2). Putting α = -1 in Theorem 2.1, we get



– 1 (G) = (12) -1

8 + (13)-1 12+(13)-1 8n+(12)-1 (4n – 4) + (14)-1 (32n – 40) + (14)-1 4n2 + (15)-1 (20n2 – 36n + 20) + (16)-1 (10n2 – 14n + 4) = 3055 2 362 6380

1560

n  n

Putting α = 2 in Theorem 2.1, we get



2 (G) = (12) 2

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ISSN(E): 2277-128X, ISSN(P): 2277-6451, pp. 96-103

+ (14)2 (32n – 40) + (14)2 4n2 + (15)2 (20n2 – 36n + 20) + (16)2 (10n2 – 14n + 4) = 1385n2 – 87n + 72.

Putting α = 1 2

 _{in Theorem 2.1, we get}

1 1 1 1 1

2 2 2 2 2

1 2

(G) (12) 8 (13) 12 (13) 8n (12) (4n 4) (14) (32n 40)

_            

1 1 1

2 2 2

(14) 4n (15) (20n 36n 20) (16) (10n 14n 4)

  

      

2 2

2 2 12 8 (20 36 20) 5 7 2 .

3 13 15 2

n  n n  n n  n

    ■

Theorem 2.3. let G = ((SK)n x n) then the S D D(G) index and IS I(G) index are (i) S D D(G)

2 2

845 7777 2533 386 1949 40797

( ) ( ) .

14 n 294 n 70 and ii ISI G 3 n 455 n 546

     

Proof:By the definition of S D D(G),

S DD(G)

2 2

( )

( ) ( ) ( ) ( )

G G

uv E G G G

d u d v d u d v









From Table 1, S DD(G)

1 2 3 4 5

74 89 85 25

2

35 E 40 E E 42 E 12 E

     2 ₆ 113 ₇ 2 ₈

56

E E E

 

5 2 1

2 2 2 ( 2)

2

r r

r

 

   

S DD(G) 845 2 7777 2533

. 14 n 294 n 70

  

By the definition of IS I(G),

IS I(G)

( )

( ) ( ) ( ) ( )

G G uv E G G G

d u d v

d u d v









From Table 1, IS I(G)

1 2 3 4 5

35 40 42 24

3

12 E 13 E E 13 E 7 E

     49 ₆ 56 ₇ ₈

4 14 E 15 E E

  

35 40 42 24

8 12 3(4 4) 8 (32 40)

12 13 n 13 n 7 n

      

2 2 2

49 56

4 (20 36 20) 4(10 14 4)

14 n 15 n n n n

      

IS I(G) 386 2 1949 40797_.

3 n 455 n 546

   ■

Theorem 2.4.The Harmonic polynomial for (S K)nxn is given by, H(G, x) = 22x11(4+6x+4(n-1) + 2nx + 4x2(4n – 5) + 2n2x2

+ 2x3(5n2 – 9n + 5)) + 2x15(5n2 – 7n + 2).

Proof:By the definition of Harmonic polynomial

H(G, x) ( ) ( ) 1

( )

2

dG u dG v

uv E G

x

 







1 1 1 1 1 1

1 2 3 4 5 6

2 1

x

E

2 2

x

E

2 1

x

E

2 2

x

E

2 3

x

E

2 3

x

E





1 1

7 8

2 4

x

E

2 6

x

E



2 11 2 2 2 3 2

2 x

(4 6

x

4(

n

1) 2

nx

4 x

(4

n

5) 2

n x

2 (5

x

n

9 n

5))





 



 







15 2

2 x

(5

n

7 n

2).







■

From the structure of 1 X 1 sodoku graph, the edge set can be patitiand into four sets show in Table 2

Table 2

(dG(u), dG(v)) where uv



E(G) Number of edges (5, 7)

(5, 8) (7, 7) (7, 8)

16 4 4 4

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ISSN(E): 2277-128X, ISSN(P): 2277-6451, pp. 96-103

Now we compute the general sum connectivity index of 1 x 1 sudoku graph.

Theorem 2.5.Consider a Sudoku graph G = (S K)1x1, Then its general sum connectivity index is



_(G) = (12)α16 + (13)α 4+ (14)α 4 + (15)α. 4

Proof:By the definition of Xα and Tabel 2, we have

, ( )

(

)

(

_G

( )

_G

( ))

v u E G

G

d

u

d

v













1 2 3 4

(12) E (13) E (14) E (15) E

   

Hence



(G)(12) 16 (13) 4 (14) 4 (15) 4. ■

Theorem 2.6.Using the above theorem, we obtain the following result, consider a Sudoku graph G = (S K)1x1, Then

360 if α = 1;

1286 , 585

if α = -1;





(G) = 4664, if α = 2;

8 4 4 4

, 2  13 14 15

if α = 1_;

2





(G) = (12)16 + (13)4 + (14)4 + (15)4 = 360.

Putting α = -1 in Theorem 2.5, we get



– 1 (G) = (12) -1

16 + (13)-14 + (14)-14 + (15)-14 = 1286_.

585

2



(G) = (12)216 + (13)24 + (14)24 + (15)24 = 4664.

Putting α = 1 2

1



_ (G) =

1 1 1 1

2 2 2 2

(12) 16 (13) 4 (14) 4 (15) 4

   

  

= 8 4 4 4 _.

2  13 14 15 ■

Theorem 2.7.For sudoku graph G = (S K)1x1 then S DD(G) index and IS I (G) index are (i) S DD(G) = 294

5

and (ii) IS

I(G) = 51426_.

586

Proof: by the definition of S DD(G) we have

S DD(G) 2 2

( )

( ) ( ) ( ) ( )

G G

uv E G G G

d u d v d u d v









From Table 2, S DD(G)

1 2 3 4

74 89 113

2

35 E 40 E E 56 E

   

S DD(G) 294. 5



IS I(G)

( )

( ) ( ) ( ) ( )

G G uv E G G G

d u d v d u d v 







From Table 2, IS I(G)

1 2 3 4

35 40 49 56

12 E 13 E 14 E 15 E

   

IS I(G) 51426_.

585

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ISSN(E): 2277-128X, ISSN(P): 2277-6451, pp. 96-103 Theorem 2.8.The Harmonic polynomial for G = (S K)1x1 is given by H(G, x) = 23x11(4+x6).

( ) ( ) 1

( )

( , ) 2 dG u dG v uv E G

H G x x  







11 12 13 14

1 2 3 4

2x E 2x E 2x E 2x E

   

3 11 6 2 x (4 x ).

  ■

III. TIO2 NANOTUBES

The molecular graph of TiO2[m, n] has total 2n + 2 rows and m columns. For TiO2 nanotubes 2 ≤ d(v) ≤ 5, for all v



V(TiO2). We denote the partitions of the vertex set of TiO2 by Vi(TiO2), where v



V(TiO2) if d(v) = i. Thus we have

the following partitions of the vertex set.

Table 3

Vertex Partition V2 V3 V4 V5 Cardinality 2mn + 4n 2mn 2n 2mn

Table 3. The vertex partition of TiO2 nanotubes

where, V2(TiO2) = {v



V(TiO2) : d(u) = 2}, V3(TiO2) = {v



V(TiO2) : d(u) = 3},

V4(TiO2) = {v



V(TiO2) : d(u) = 41} and V5(TiO2) = {v



V(TiO2) : d(u) = 5}.

Again the edge set of TiO2 is divided into three edge partitions based on the sum of degrees of the end vertices and denote it by Ej(TiO2) so that if e = uv



Ej (TiO2) then d(u)+d(v) = j for



(G) ≤ j ≤ ∆ (G). Thus we write,

E(TiO2) =

2 ( ),

j

Ej TiO

 





where E6 (TiO2) = {e = uv



E(TiO2) : d(u) = 2 & d(v) = 4},E7(TiO2) =

E

₇'∪

E

₇"

,

where E’7 = {e = uv



E(TiO2) : d(u) = 2 & d(v) = 5} and

" 7

E

= {e = uv



E(TiO2) : d(u) = 3 & d(v) = 4}.

E8(TiO2) = {e = uv



E(TiO2) : d(u) = 3 & d(v) = 5}. From direct calculation, we get |E6(TiO2)| = 6n, |E2(TiO2)| = 4mn + 4n and |E8(TiO2)| = 6mn – 2n.

Similarly, the edge set of TiO2 is also divided into four edge partitions based on the product of degrees of the end vertices and denote it by (TiO2) so that if then d(u)d(v) = k for δ(G)2 ≤ k ≤ ∆(G)2. Thus we have the following partitions of the edge set.

(TiO2) = {e = uv



E (TiO2) : d(u) = 2 & d(v) = 4}, (TiO2) = {e = uv



E (TiO2) : d(u) = 2 & d(v) = 5}, (TiO2) = {e = uv



E (TiO2) : d(u) = 3 & d(v) = 4} and (TiO2) = {e = uv



E (TiO2) : d(u) = 3 & d(v) = 5}.

Table 4

Edge Partition E6 = E7 E8 = Cardinality 6n 4mn + 4n 6mn -2n 4mn +2n 2n

Table 4. The edge partition of TiO2 nanotubes

From the structure of TiO2 nanotubes, we get | (TiO2)| = 6n, | (TiO2)| = 4mn + 2n, | (TiO2)| = 2n and | (TiO2)| = 6mn – 2n. Clearly, E7 = , E6 = , E8 = . The partitions of the edge set of TiO2 nanotubes is given in Table 4. In the following we calculate the general sum-connectivity index, atom-bond connectivity index, geometric arithmetic index, inverse sum indeg index, symmetric division deg index and harmonic polynomial TiO2[m, n] nanotube as defined in the previous section.

Theorem 3.1.General sum connectivity index of TiO2 is given by Xα(G) = (6)α 6n + (7)α (4mn + 12n) + (7)α 2n + (8)α 6mn

– 2n.

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ISSN(E): 2277-128X, ISSN(P): 2277-6451, pp. 96-103

 

, ( )

(

_G

( )

_G

( ))

v u E G

d

u

d

v

G













* * * *

8 10 12 15

(6) E (7) E (7) E (8) E

   

Hence _(G)(6) 6 n(7) (4 mn12 )n (7) 2 n(8) .6 mn2 .n■

Theorem 3.2.Using the above theorem, we obtain the following result, consider G = TiO2, Then

4(19mn + 12n) if α = 1;

37 37 , 28

mn n _{if α =}_-1;





(G) = 580mn + 284n, if α = 2;

6 4 14 3

,

6 7 2

n _ mn n_ mnn _ifα = 1_;

2





( )G (6)6n7(4mn2 )n (7)2n8(6mn2 )n 4(19mn12 ).n

Putting α = -1 in Theorem 2.9, we get

1 1 1 1

1( )G (6) 6n (7) (4mn 12 )n (7) 2n (8) .6mn 2n



   

      

37 37

28

mn n





₂

( )

G



(6) 6

2

n



(7) (4

2

mn



12 )

n



(7) 2

2

n



(8) .6

2

mn



2 n

580mn284 .n

Putting α = 1 2

1 1 1 1

2 2 2 2

1(G) (6) 6n (7) (4mn 12 )n (7) 2n (8) .6mn 2n

          

6 4 4 3 _.

6 7 2

n mn n mnn

   ■

Theorem 3.3. For G = TiO2, we have

8 4 5 6 2

( ) ( ) .

2 3 2

5

n mn n mn n

i ABC G     

2 15

( ) ( ) 4 2 10(4 2 ) 12(2 ) 3 .

7 2

ii GA G  n  mn n  n  mnn

756 613

( ) ( ) .

30

mn n iii SDD G  

475 295

( ) ( ) .

28

mn n vi ISI G  

Proof. By the definition of atom-bond connectivity index,

( )

( ) ( ) 2 ( )

( ) ( )

G G

uv E G G G

d u d v ABC G

d u d v



 





* *

8( ) 10

( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) ( )

G G G G

uv E G G G uv E G G

d u d v d u d v d u d v d u d v

 

   









* *

12( ) 15

( )

2 ( )

( )

2 ( )

( )

G G G G

uv E G G G uv E G G

d

u

d

v

d

u

d

v

d

u d

v

d

u d

v

 

















8 4 5 6 2

.

2 3 2

5

n mn n mn n

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ISSN(E): 2277-128X, ISSN(P): 2277-6451, pp. 96-103

By the definition of Geometric Arithmetic index,

( )

2 ( ) ( ) ( )

( ) ( )

G G

uv E G G G

d u d v GA G

d u d v









* *

8 10

2 ( )

( )

2 ( )

( )

G G G G

uv E G G uv E G G

d

u d

v

d

u d

v

d

u

d

v

d

u

d

v

 







* *

12 15

2 ( )

( )

2 ( )

( )

G G G G

d

u d

v

d

u d

v

d

u

d

v

d

u

d

v

 





4

2

2 ( 10(4

2 )

122

15

3 .

7

2 n

mn

n

mn

n







Similarly we can prove (iii) and (vi) ■

Theorem 3.4.The Harmonic polynomial for G = TiO2is given by H(G, x) = 22x5 (3n + x(2mn + n) + xn + x2 (3mn – n)).

( ) ( ) 1

( )

( , ) 2 dG u dG v uv E G

H G x x  









2 x E

5 ₈*



2 x

6

E

₁₀*



2 x

6

E

₁₂*



2 x

7

E

₁₅*

2 5 2

2 x (3n x(2mn n) xn x (3mn n)).

      ■

Table 4

Edge Partition E6 E7 E8 Cardinality 6n 4mn + 4n 6mn -2n

Theorem 3.5.For G = TiO2, we have

2(5 2 ) 56 2

( ) ( ) .

2 2

5

n mn mn n

i ABC G    

2 2 8 10 15 15

( ) ( ) ( 10( ) 2 3 .

3 7 2 2

ii GA G   mnn  n mnn

276 365

( ) ( ) .

15

mn n iii SDD G  

475 279

( ) ( ) .

28

mn n

vi ISI G  

Proof.By the definition of atom-bond connectivity index,

( )

2 ( )

( )

G G

uv E G G G

d

u

d

v

ABC G

d

u d

v











6 7

( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) ( )

G G G G

d u d v d u d v d u d v d u d v

 

   









8

( )

2 ( )

( )

G G

uv E G G

d

u

d

v

d

u d

v











2(5 2 ) 56 2 .

2 2

5

n mn mn n

 

By the definition of Geometric Arithmetic index,

( )

2 ( )

( )

(

)

( )

G G

uv E G G G

d

u d

v

GA G

d

u

d

v









6 7 8

2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

G G G G G G

uv E G G uv E G G uv E G G

d u d v d u d v d u d v d u d v d u d v d u d v

  

  

  



2 2 8 10 15 15

( 10 ( )) 2 3 .

3 7 mn n 2 n 2 mn n

(8)

ISSN(E): 2277-128X, ISSN(P): 2277-6451, pp. 96-103

Similarly we can prove (iii) and (vi)

Theorem 3.6.The Harmonic polynomial for G = TiO2 is given by H(G, x) = 22x5(3n+2x(mn + n) + x2 (3mn – n)).

( ) ( ) 1

( )

( , ) 2 dG u dG v uv E G

H G x x  







5 6 7

6 7 8

2x E 2x E 2x E

  



2

x

5

(3

n



2 (

x mn



n

)



x

2

(3

mn



n

)).

■

IV. CONCLUSION

The study of the degree based topological indices for important classes of graphs is an essential problem in Chemical graph theory. In this view, we obtain the same topological indices of Titania TiO2 Nanotubes and Sudoku

graphs.

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