• No results found

SD-Prime Cordial Labeling of Subdivision of Snake Graphs

N/A
N/A
Protected

Academic year: 2020

Share "SD-Prime Cordial Labeling of Subdivision of Snake Graphs"

Copied!
10
0
0

Loading.... (view fulltext now)

Full text

(1)

IJSRR, 8(2) April. – June., 2019 Page 2414

Research article Available online www.ijsrr.org

ISSN: 2279–0543

International Journal of Scientific Research and Reviews

SD-Prime Cordial Labeling of Subdivision of Snake Graphs

U. M. Prajapati

1*

and A. V. Vantiya

2

1

St. Xavier’s College, Ahmedabad, India. Email: udayan64@yahoo.com

2

Research Scholar, Department of Mathematics,Gujarat University, K. K. Shah Jarodwala Maninagar Science College, Ahmedabad, India.

Email: avantiya@yahoo.co.in

ABSTRACT:

Let ∶ ( )→{1, 2, … , | ( )|} be a bijection, and let us denote = ( ) + ( ) and

= | ( )− ( )| for every edge in ( ). Let ′ be the induced edge labeling, induced by the

vertex labeling , defined as : ( )→ {0,1} such that for any edge in ( ), ( ) = 1 if

( , ) = 1, and ( ) = 0 otherwise. Let (0) and (1) be the number of edges labeled

with 0 and 1 respectively. Then is said to be SD-prime cordial labeling if (0)− (1) ≤ 1

and is said to be SD-prime cordial graph if it admits SD-prime cordial labeling. In this paper, we

investigate the SD-prime cordial labeling behaviour of subdivision of some snake graphs, namely

subdivision of: triangular snake, alternate triangular snake, quadrilateral snake, alternate quadrilateral

snake.

AMS Subject Classification (2010): 05C78.

KEYWORDS:

SD-prime cordial graph, triangular snake, subdivision of alternate triangular snake,

subdivision of quadrilateral snake, subdivision of alternate quadrilateral snake.

*Corresponding Author:

U. M. Prajapati

St. Xavier’s College,

Ahmedabad - 380009, India.

(2)

IJSRR, 8(2) April. – June., 2019 Page 2415

INTRODUCTION:

Let = ( ), ( ) be a simple, finite and undirected graph of order | ( )| = and size

| ( )| = . For standard terminology of Graph Theory, we used1. For all detailed survey of graph

labeling we refer2. Lau, Chu, Suhadak, Foo, and Ng3 have introduced SD-prime cordial labeling and

they proved behaviour of several graphs like path, complete bipartite graph, star, double star, wheel,

fan, double fan and ladder. Lourdusamy and Patrick4 proved that

, , , , , , , , , , , , , , , , , , ⋆

, , , , , ( ), ( ), , ( ), , ⊙ and ⊙ the graph obtained by

duplication of each vertex of path and cycle by an edge are SD-prime cordial. Lourdusamy, Wency

and Patrick5 proved that the union of star and path graphs, subdivision of comb graph, subdivision of

ladder graph and the graph obtained by attaching star graph at one end of the path are SD-prime

cordial graphs. They proved that the union of two SD-prime cordial graphs need not be SD-prime

cordial graph. Also, they proved that given a positive integer , there is SD-prime cordial graph

with vertices. Prajapati and Vantiya6 proved that ( ≠3), ( ), , ( ), , ( ),

and ( ) are SD-prime cordial.

Definition 1:If the vertices or edges or both of a graph are assigned values subject to certain

conditions then it is known as vertex or edge or total labeling respectively.

Definition 2:3 A bijection : ( )→{1, 2, … , | ( )|} induces an edge labeling : ( )→ {0, 1}

such that for any edge in , ( ) = 1 if ( , ) = 1, and ′( ) = 0 otherwise, where

= ( ) + ( ) and = | ( )− ( )| for every edge in ( ). The labeling is called

SD-prime cordial labeling if (0)− (1) ≤ 1. is called prime cordial graph if it admits SD-prime cordial labeling.

Definition 3:1 The subdivision graph ( ) is obtained from by subdividing each edge of by a

vertex.

Definition 4:2 A triangular snake is obtained from the path by replacing every edge of a path

by a triangle . That is, it is obtained from a path , , … , by joining and to a new

vertex for = 1, 2, … , −1.

Definition 5:2 An alternate triangular snake ( ) is obtained from the path by replacing every

alternate edge of a path by a triangle . That is, it is obtained from a path , , … , by joining

(3)

IJSRR, 8(2) April. – June., 2019 Page 2416

Definition 6:2 An alternate quadrilateral snake ( ) is obtained from the path = , , … ,

by replacing every alternate edge of a path by a cycle , in such a way that each pair of vertices

( , ) remains adjacent. That is, it is obtained from a path = , , … , by joining and

(alternately) to new vertices and respectively, and then joining and by an edge, for

= 1, 2, … , −1.

Notation: Throughout this paper, a path = , , … , , where > 1.

MAIN RESULTS:

Theorem 1:The graph ( ) is SD-prime cordial.

Proof:Let ( ) = ( )∪ { , , , :1≤ ≤ −1} and

( ) = { , , , , , ∶ 1≤ ≤ −1}. Therefore, ( ) is of order

5 −4 and size 6 −6.

Figure 1: ( )

Define : ( ) →{1, 2, … , 5 −4} as follows:

( ) = 5 −4 1≤ ≤ ;

( ) = 5

5 −1

≢0( 3), 1≤ ≤ −1;

≡0( 3), 1≤ ≤ −1;

( ) = 5 −3

5

≢0( 3), 1≤ ≤ −1;

≡0( 3), 1≤ ≤ −1;

( ′) = 5 −1

5 −3

≢0( 3), 1≤ ≤ −1;

≡0( 3), 1≤ ≤ −1;

( ′′) = 5 −2 1≤ ≤ −1.

Therefore, (0) = (1) = 3 −3.

Thus (0)− (1) ≤1. Hence ( ) is SD-prime cordial.

Theorem 2:The graph ( ) is SD-prime cordial.

Proof:

Case-1: Let the first triangle be starts from and the last triangle be ends at :

(4)

IJSRR, 8(2) April. – June., 2019 Page 2417

Let ( ) = ( )∪{ :1≤ ≤ −1}∪{ , , : 1≤ ≤ −1}

and ( ) = { , :1≤ ≤ −1}∪{ , , , : 1≤ ≤ −1}. Therefore, in this case, ( ) is of order and size 4 −2.

Figure 2: ( )

Define : ( ) → 1, 2, … , as follows:

( ) = 14 −9 + (−1)

4 1≤ ≤ ;

( ) = 14 −3 + 3(−1)

4 1≤ ≤ −1;

( ) =7 + 5

2 1≤ ≤ −1;

( ′) =7 + 1

2 1≤ ≤ −1;

( ′′) =7 −1

2 1≤ ≤ −1.

Therefore, if (0) = (1) = 2n−1.

Case-2: Let the first triangle be starts from and the last triangle be ends at .

In this case, will be an odd number. Let > 1. Define ( ) and ( ) as per

Case-1. Therefore, in this case, ( ) is of order and size 4 −4.

Figure 3: ( )

(5)

IJSRR, 8(2) April. – June., 2019 Page 2418

Remark: Note that, the graphs of case-2 and case-3 are isomorphic graphs, so it is enough to prove

that any one of these two graphs is SD-prime cordial. But here we have given separate labelings for

both the cases.

Case-3: Let the first triangle be starts from and the last triangle be ends at .

In this case, will be an odd number. Let > 2.

Let ( ) = ( )∪{ :1≤ ≤ −1}∪{ , , : 1≤ ≤ −

1}and ( ) = { , :1≤ ≤ −1}∪{ , , , :

1≤ ≤ −1}.Therefore, in this case, ( ) is of order and size 4 −4.

Figure 4: ( )

Define : ( ) → 1, 2, … , as follows:

( ) = 2, ( ) = 1,

( ) =14 −15−(−1)

4 2≤ ≤ ;

( ) =14 −9−3(−1)

4 2≤ ≤ −1;

( ) =7 + 2

2 1≤ ≤ −1;

( ′) =7 −2

2 1≤ ≤ −1;

( ′′) =7 −4

2 1≤ ≤ −1.

Therefore, (0) = (1) = 2n−2.

Case-4:Let the first triangle be starts from and the last triangle be ends at .

In this case, will be an even number. Let > 2. Define ( ) and ( ) as per

(6)

IJSRR, 8(2) April. – June., 2019 Page 2419

Figure 5: ( )

Define : ( ) → 1, 2, … , as per the Case-3. Therefore, (0) = (1) = 2n−3.

Thus in all cases (0)− (1) ≤1. Hence ( ) is SD-prime cordial.

Theorem 3:The graph ( ) is SD-prime cordial.

Proof:Let ( ) = ( )∪ { , , , , , :1≤ ≤ −1} and

( ) = { , , , , , , , :1≤ ≤ −1}. Therefore, ( )

is of order 7 −6 and size 8 −8.

Figure 6: ( )

Define : ( ) →{1, 2, … , 7 −6} as follows:

( ) = 7 −6 1≤ ≤ ;

( ) = 7 1≤ ≤ −1;

( ) = 7 −5 1≤ ≤ −1;

( ) = 7 −2 1≤ ≤ −1;

( ) = 7 −4 1≤ ≤ −1;

( ) = 7 −1 1≤ ≤ −1;

( ) = 7 −3 1≤ ≤ −1.

Therefore, (0) = (1) = 4n−4.

Thus (0)− (1) ≤1. Hence ( )is SD-prime cordial.

Theorem 4:The graph ( ) is SD-prime cordial.

Proof:Case-1: Let the first cycle be starts from and the last cycle be ends at .

(7)

IJSRR, 8(2) April. – June., 2019 Page 2420

Let ( ) = ( )∪ { :1 ≤ ≤ −1}∪ { , , , , : 1≤ ≤ −

1} and ( ) = { , :1≤ ≤ −1}∪{ , , , , ,

: 1≤ ≤ −1}. Therefore, in this case, ( ) is of order and size

5 −2.

Figure 7: ( ) ∶ −

Define : ( ) → 1, 2, … , as follows:

( ) =

⎩ ⎨

⎧18 −9 + 5(−1)

4

18 −13 + 5(−1)

4

≡1( 4) ≡2( 4) 1≤ ≤ ;

≡3( 4) ≡0( 4) 1≤ ≤ ;

( ) =

⎩ ⎨

⎧18 + 7−3(−1)

4

18 + 7−7(−1)

4

≡1( 4) ≡2( 4) 1≤ ≤ −1;

≡3( 4) ≡0( 4) 1≤ ≤ −1;

( ) =

9 −5

2

9 −3

2

≡1( 4) 1≤ ≤ −1;

≡3( 4) 1≤ ≤ −1;

( ) =

9 + 1 2

9 −1

2

≡1( 4) 1≤ ≤ −1;

≡3( 4) 1≤ ≤ −1;

( ) =

9 −3

2

9 −5

2

≡1( 4) 1≤ ≤ −1;

≡3( 4) 1≤ ≤ −1;

( ′) =9 + 3

2 1≤ ≤ −1;

( ) =

9 −1

2 9 + 1

2

≡1( 4) 1≤ ≤ −1;

≡3( 4) 1 ≤ ≤ −1.

(8)

IJSRR, 8(2) April. – June., 2019 Page 2421

Case-2: Let the first cycle be starts from and the last cycle be ends at .

In this case, will be an odd number. Let > 1. Define ( ) and ( ) as per the

Case-1. Therefore, in this case, ( ) is of order and size 5 −5.

Figure 8: ( ) ∶ −

Define : ( ) → 1, 2, … , asper the case-1.

Therefore, (0) = (1) = .

Case-3: Let the first cycle be starts from and the last cycle be ends at .

In this case, will be an odd number. Let > 2.

Let ( ) = ( )∪ { :1 ≤ ≤ −1}∪ { , , , , : 1≤ ≤

−1} and ( ) = { , :1≤ ≤ −1}∪{ , , , , ,

: 1≤ ≤ −1}. Therefore, in this case, ( ) is of order and size

5 −5.

Figure 9: ( ) ∶ −

(9)

IJSRR, 8(2) April. – June., 2019 Page 2422 ( ) =

⎩ ⎨

⎧18 −21−7(−1)

4

18 −21−3(−1)

4

≡1( 4) ≡2( 4) 1≤ ≤ ;

≡3( 4) ≡0( 4) 1≤ ≤ ;

( ) =

⎩ ⎨

⎧18 −1 + 5(−1)

4

18 −17−7(−1)

4

≡1( 4) ≡2( 4) 1≤ ≤ −1;

≡3( 4) ≡0( 4) 1≤ ≤ −1;

( ) =

9 −8

2

9 −10

2

≡2( 4) 1≤ ≤ −1;

≡0( 4) 1≤ ≤ −1;

( ) =

9 −6

2

9 −4

2

≡2( 4) 1≤ ≤ −1;

≡0( 4) 1≤ ≤ −1;

( ) =

9 −10

2

9 −8

2

≡2( 4) 1≤ ≤ −1;

≡0( 4) 1≤ ≤ −1;

( ′) =9 −2

2 1≤ ≤ −1;

( ′′) =

9 −4

2

9 −6

2

≡2( 4) 1≤ ≤ −1;

≡0( 4) 1≤ ≤ −1.

Therefore, (0) = (1) = .

Case-4: Let the first cycle be starts from and the last cycle be ends at .

In this case, will be an even number. Let > 2. Define ( ) and ( ) as per

the Case-3. Therefore, in this case, ( ) is of order and size 5 −8.

Figure 10: ( ) ∶ −

Define : ( ) → 1, 2, … , as per the Case-3.

Therefore, (0) = (1) = .

(10)

IJSRR, 8(2) April. – June., 2019 Page 2423

CONCLUSION:

We have proved that the graphs ( ), ( ) , ( ) and ( ) are SD-prime cordial.

Further investigation can be done for subdivision of other graph family.

REFERENCES:

1. Bondy J. A. and Murty U. S. R.,Graph theory with applications, 1st ed, New York,

MacMillan, 1976; 1-651.

2. Gallian J. A., A Dynamic Survey of Graph Labeling, The Electronic Journal of

Combinatories, 2018; 1-502.

3. Lau G. C., Chu H. H., Suhadak N., Foo F. Y. and Ng H. K., On SD-Prime Cordial Graphs,

International Journal of Pure and Applied Mathematics, 2016; 106(4): 1017-1028.

4. Lourdusamy A. and Patrick F., Some Results on SD-Prime Cordial Labeling, Proyecciones

Journal of Mathematics, 2017; 36(4): 601-614.

5. Lourdusamy A., Wency S. Jenifer and Patrick F., On SD-Prime Cordial Labeling,

International Journal of Pure and Applied Mathematics, 2017;117(11): 221-228.

6. Prajapati U. M. and Vantiya A. V.,SD-Prime Cordial Labeling of Some Snake Graphs,

Figure

Figure 1:�(��)
Figure 2:���(��)�
Figure 4:���(��)�
Figure 5:���(��)�
+4

References

Related documents

Consistent with theoretical predictions for populations experiencing genetic drift via founder events or population bottlenecks (Nei et al. 1975; Novak and Mack 2005), I have

In conclusion, after obtaining data from a randomized distribution of TOMS Shoes in rural communities of El Salvador, I studied the effects of shoe distribution on time

This conforms well with our prediction that island area would be positively correlated with genetic diversity and suggests that larger islands support a larger population size and

Conclusion: Infertility was due to various causes in the male or/and the female partner, however, infertility among couples were mostly due to the problem in females.. Infertility

After applying the face detection and feature extraction to all images in the dataset, a dataset containing all features of all faces in the dataset has been

Habitat of Mesoclemmys vanderhaegei observed at the Ecological Station of Santa Barbara, Águas de Santa Bárbara municipality, state of São Paulo, southeastern Brazil..

Electronic technology Inter Integrated Circuit (I2C) temperature and humidity sensors SHT11 is a single chip temperature and relative humidity sensor with multi

Based on the topology of a 3D data bus with a number of timing periods for accessing data, an effective algorithm is proposed to insert signal repeaters into the critical