DIVIDED SQUARE DIFFERENCE CORDIAL LABELING OF SPLITTING GRAPHS

International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 9, Issue 2, March April 2018, pp. 87 93, Article ID: IJARET_09_02_011 Available online at http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=9&itype=2 ISSN Print: 0976-6480 and ISSN Online: 0976-6499 IAEME Publication DIVIDED SQUARE DIFFERENCE CORDIAL LABELING OF SPLITTING GRAPHS A. Alfred Leo Research Scholar, Research and Development center, Bharathiar University, Coimbatore, Tamil Nadu, India R. Vikramaprasad Assistant Professor, Department of Mathematics, Government Arts College, Salem, Tamil Nadu, India ABSTRACT In this paper, we investigate the concepts of splitting graph of divided square difference cordial labeling behavior of path, cycle,,,,,,,. Keywords: Splitting graphs, Divided square difference cordial labeling,, ;, ; + ; +2. Cite this Article: A. Alfred Leo and R. Vikramaprasad, Divided Square Difference Cordial Labeling of Splitting Graphs International Journal of Advanced Research in Engineering and Technology, 9(2), 2018, pp 87 93. http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=9&itype=2 1. INTRODUCTION All graphs in this paper are finite, simple and undirected. We use the basic notation and terminology of graph theory as in [7], while for number theory we refer Burton [2] and graph labeling as in [6]. Most graph labeling methods were introduced by Rosa [10] in 1967. The concept of cordial labeling was introduced by Cahit [1]. Alfred Leo et.al [5] introduced the concept of divided square difference cordial labeling graphs. S.Abhirami et.al[4] investigated the concept of Splitting graph on even sum cordial labeling graphs. In this paper, we investigate the concepts of divided square difference cordial labeling of splitting graphs for path, cycle,, ;, ;,,. 2. PRELIMINARIES 2.1. Definition The Graph labeling is an assignment of numbers to the edges or vertices or both subject to certain condition(s). If the domain of the mapping is the set of vertices (edges), then the labeling is called a vertex (edge) labeling. Graph labeling has large number of applications in Mathematics as well as in several areas of Computer science and Communication networks. For a dynamic survey on various graph labeling problems we refer to Gallian [6]. http://www.iaeme.com/ijaret/index.asp 87 editor@iaeme.com

Divided Square Difference Cordial Labeling of Splitting Graphs 2.2. Definition A mapping : 0,1 is called binary vertex labeling of G and is called the label of the vertex of G under. The concept of cordial labeling was introduced by Cahit [1]. 2.3. Definition A binary vertex labeling f of a graph G is called a Cordial labeling if 0 1 1 and 0 1 1. A graph G is cordial if it admits cordial labeling. 2.4. Definition (Splitting Graph) For each vertex of a graph, take a new vertex, join to all vertices of G adjacent to. The graph thus obtained is called splitting graph of. 2.5. Definition Let =, be a simple graph and : 1,2,3, be bijection. For each edge ", assign the label 1 if # $% &' % # is odd and the label 0 otherwise. f is called Divided $&' square difference Cordial Labeling if 0 1 1, where 1 and 0 denote the number of edges labeled with 1 and not labeled with 1 respectively. 2.6. Proposition 1. Any Path is a divided square difference cordial graph. 2. Any Cycle + is a Divided square difference cordial graph except 3.,=6,6+.,6+2., When.=4. 4. The Star graph, is a divided square difference cordial graph. 5. The complete bipartite graph, is a divided square difference cordial graph. 6. The graph + is a divided square difference cordial. 7. The double fan +2 is a divided square difference cordial graph. 3. MAIN RESULT 3.1. Proposition The graph when n is even is a divided square difference cordial graph. Let G be a graph (,). Let,,, are the vertices of the path. Then,,,,,,, are the vertices of. In the graph, =2, and =3, 1. Define a map 1,2,,2,. By Proposition 2.6, we can construct the divided square difference cordial path. Then draw the splitting graph and label the graph by taking &8 =,+9+1,0 9, 1. Thus 0 1 1. http://www.iaeme.com/ijaret/index.asp 88 editor@iaeme.com

A. Alfred Leo and R. Vikramaprasad 3.2. Example : 3.3. Proposition The graph + when n is even is a divided square difference cordial graph except,= 6,6+.,6+2., when.=4. Let G be a graph +. Let,,, are the vertices of the cycle +. then,,,,,,, are the vertices of +. In the graph +, =2, and =3,. Define a map :+ 1,2,,2,. By Proposition 2.6, we can construct the divided square difference cordial Cycle +. Then by introducing new vertices, we can form the splitting graph +. We can label the splitting graph + by taking 8 =,+9,1 9,. In the resultant graph, 0 1 1. 3.4. Example + B 3.5. Proposition The Star graph C, D is a divided square difference cordial graph. Let G be a graph C, D. Let E,,,, are the vertices of the graph,. then E,,,,,E F,,,, are the vertices of C, D. In the graph C, D, =2,+2 and =3,. http://www.iaeme.com/ijaret/index.asp 89 editor@iaeme.com

Divided Square Difference Cordial Labeling of Splitting Graphs Define a map : 1,2,,2,2. Now we can construct C, D as follows: E 1, 8 92,19, 1 E 2, G,H2,1H,. 2 In the resultant graph we get 0, if, is even 1 0I 1, if, is odd Thus 0 11. 3 3.6 Example 1. When m is odd (m = 3) LCM N,O D 2. When m is even (m = 4) 3.7. Proposition LCM N,O D The graph C, D is a divided square difference cordial graph. Let G be a graph C, D. Let E,E,,,, are the vertices of the graph,. then E,E,,,,,E F,E F,,,, are the vertices of C, D. In the graph C, D, 2,4 and 6,. Define a map : 1,2,,2,4. http://www.iaeme.com/ijaret/index.asp 90 editor@iaeme.com

A. Alfred Leo and R. Vikramaprasad Now we can label C, D as follows: E =1,E =2, 8 =9+2, 1 9, 4 E =,+3,E =,+4, G =,+H+4, 1 H,. 5 We get, 1= 0=3, for every n. Thus 0 1 1. 3.8. Example 3.9. Proposition LCM P,O D The graph C,, D is a divided square difference cordial graph. Let G be a graph C,, D. Let E,,,,, Q, Q,, are the vertices of the graph,,. Then E,,,,, Q, Q,,,E F, F, F,, F, Q, Q,, are the vertices of C,, D. In the graph C,, D, =4,+2 and =6,. Define a map : 1,2,,4,+2. Now we can label the vertices of C,, D as follows: E =1, 8 =29, 1 9 2, 6 E =4,+2, Q&8 =29 1,,+2 9 2,+1 7 &8 =29+3, 0 9, 1. 8 In this graph we get, 1= 0 for every n. Thus 0 1 1. http://www.iaeme.com/ijaret/index.asp 91 editor@iaeme.com

3.10. Example 1. When n is even Divided Square Difference Cordial Labeling of Splitting Graphs 2. When n is odd SCT N,P,P D LCM N,O,O D 4. CONCLUSION In this paper, the concepts of divided square difference cordial labeling behavior of splitting graphs of path, cycle,,,,,,, were discussed. REFERENCES 1. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars Combinatoria, 23 (1987), pp. 201 207. 2. David M. Burton, Elementary Number Theory, Second Edition, Wm. C. Brown Company Publishers, 1980. 3. R.Dhavaseelan, R.Vikramaprasad and S.Abhirami; A New Notions of Cordial Labeling Graphs, Global Journal Of Pure And Applied Mathematics, 11(4)(2015), pp.1767 1774. 4 R.Vikramaprasad, R.Dhavaseelan and S.Abhirami; Splitting graph on even sum cordial labeling graphs, International journal of Mathematical archive, 7(3),2016, pp.91-96. 5. A.Alfred Leo, R.Vikramaprasad and R.Dhavaseelan; Divided square difference cordial labeling graphs, International journal of Mechanical Engineering and Technology, 9(1), Jan 2018, pp.1137 1144. http://www.iaeme.com/ijaret/index.asp 92 editor@iaeme.com

A. Alfred Leo and R. Vikramaprasad 6. J. A. Gallian, A dynamic survey of graphs labeling, Electronic J. Combin. 15(2008), DS6, pp.1 190. 7. F. Harary, Graph theory, Addison-Wesley, Reading, MA. (1969). 8. P.Lawrence Rozario Raj and P.Lawrence Joseph Manoharan, Some results on divisor cordial labeling of graphs, International Journal of Innovative Science, Engineering & Technology, Vol.1, Issue10,2014, pp.226-231. 9. P. Lawrence Rozario Raj and R. Valli, Some new families of divisor cordial graphs, International Journal of Mathematics Trends and Technology, 7(2), 2014, pp.94 102. 10. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Paris (1967), pp.349 355. 11. S.K.Vaidya and N.H.Shah, Further results on divisor cordial labeling, Annals of Pure and Applied Mathematics, 14(2), 2013, pp. 150-159. http://www.iaeme.com/ijaret/index.asp 93 editor@iaeme.com