SOME EXTENSION OF 1-NEAR MEAN CORDIAL LABELING OF GRAPHS. Sattur , TN, India.

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1 SOME EXTENSION OF 1-NEAR MEAN CORDIAL LABELING OF GRAPHS A.Raja Rajeswari 1 and S.Saravana Kumar 1 Research Scholar, Department of Mathematics, Sri S.R.N.M.College, Sattur-66 03, TN, India. Department of Mathematics, Sri S.R.N.M.College, Sattur-66 03, TN, India. ABSTRACT : Let G = (V, E) be a simple graph. A bijective function f: V(G) {0, 1,..., p 1}is said to be a 1-Near Mean Cordial Labeling if for each edge uv, the induced map f*(uv) = 0 if f u +f v is an integer 1 oterwise Satisfies the condition e f (0) e f (1) 1 where e f (0) is the number of edges with 0 label and e f (1) is the number of edges with 1 label. G is said to be a 1-Near Mean Cordial Graph if it has a 1- Near Mean Cordial Labeling. In this paper,we proved that path, cycle, complete bipartite, star, fan, crown, comb and D (P n ) are 1- Near Mean Cordial Graphs. Keywords : 1-Near Mean Cordial Labeling, 1-Near AMS Subject Classification (010): 05C78 1 INTRODUCTION All graphs considered here are finite, simple and undirected. Gallian [] has given a dynamic survey of labeling. For graph theoretic terminologies and notations we follow Harary [3]. The concept of mean cordial labeling was introduced by Raja Ponraj, Muthirulan Sivakumar and Murugesan Sundaram in the year 01 [1,4,5,7]. Let f be a function from V(G) to {0,1,}. For each edge uv of G, assign the f(u)+f(v) label. f is called a mean cordial labeling of G if v f (0) v f (1) 1 and e f (0) e f (1) 1, i, j {0,1,} where v f (x) and e f (x) denote the number of vertices and edges labeled with x (x=0,1,) respectively. A graph with a mean cordial labeling is called Mean Graph. K. Palani, J. Rejila Jeya Surya [6] introduced a new concept called 1-Near Mean Cordial Labeling and investigated some standard graphs. PRELIMINARIES We define the concept of 1-Near Mean Cordial Labeling as follows, Let G = (V, E) be a simple graph. A bijective function f: V {0, 1,..., p 1} said to be a 1-Near Mean Cordial Labeling if for each edge uv, the induced map f*(uv) = 0 if f u +f v is an integer 1 oterwise Satisfies the condition e f (0) e f (1) 1 where e f (0) is the number of edges with 0 label and e f (1) is the number of edges with 1 label. G is said to be a 1-Near Mean Cordial Graph if it has a 1- Near Mean Cordial Labeling. Definition.1. If all the vertices in a walk are distinct, then it is called a path and a path of length k is denoted by P k+1. ISSN: All Rights Reserved 016 IJSETR 653

2 Definition.. A cycle is a circuit in which no vertex except the first (which is also the last) appears more than once. Alternatively, a cycle can be defined as a closed path. Definition.3. A graph G is called a complete bipartite graph K m,n with bipartition V(G) = V 1 V where V 1 = {x 1, x,..., x m } and V = {y 1, y,..., y n } and all vertices in V 1 are adjacent to all vertices in V but no vertices in V 1 and V. Definition.4. The graph K 1,r, r 1 is called a star at the vertex has degree r is called center. Definition.5. The join G 1 + G of G 1 and G consists of G 1 G and all lines joining V 1 with V as vertex set V (G 1 ) V (G ) and edge set E[G 1 G ] = E(G 1 ) E(G ) [uv : u V (G 1 )and v V (G )]. The graph P n + K 1 is called a fan. Definition.6. The crown (C n ʘ K 1 ) is obtained by joining a pendant edge to each vertex of C n. Definition.7. The corona (G 1 ʘ G ) of two graphs G 1 and G is defined as the graph G obtained by taking one copy of G 1 (which has p 1 points) and p 1 copies of G and then joining the i th point of G 1 to every point in the i th copy of G. The graph P n ʘ K 1 is called a comb. Let G be P n. Let V(G) = {u i :1 i n} and E(G) = {(u i u i+1 ) : 1 i n 1} Case:(i) when f(u i+1 ) = i, 0 i n 1 f(u i ) = n i, 1 i n, 0 i 0 mod 1 i 1 mod 1 i n 1 Here e f (0) = n 1, e f (1) = n Case:(ii) when f(u i+1 ) = i, 0 i n 1 f(u i ) = n i, 1 i n 1 0 i 1 mod 1 i 0 mod 1 i n 1 Here e f (0) = e f (1) = n 1 Therefore, a path P n is a 1-near mean cordial graph. Illustration 1. The 1-near mean cordial graph of P 8 and P 7 are shown in the figure 1(a) and 1(b) Definition.8. D (P n ) is the graph of two copies of path graph P n consisting of vertices u i and v i for 1 i n. Hence D (P n ) consists of V(G)=n and E(G)=4(n-1). 3 MAIN RESULTS Theorem 3.1. A path P n is a 1-Near Mean Cordial Graph. Proof. Let G=(V, E) be a simple graph. ISSN: All Rights Reserved 016 IJSETR 654

3 Theorem 3.. A cycle C n is a 1-Near Mean Cordial Graph for n mod 4, n 3. Let G be C n Let V (G) = {u i : 0 i n 1} and E(G)={[(u i u i+1 ) : 1 i n ] (u n 1 u 0 )} Case(i): when n 0 mod 4 f(u i 1 ) = n i, 1 i n f(u i ) = i, 0 i n 1 0 i 1 mod 1 i 0 mod 0 i n f*(u n 1 u 0 ) = 0 Here e f (0) = e f (1) = n Therefore, the graph, C n, is a 1-near mean f*(u n 1 u 0 ) = 0 Here, e f (0) = 0 i 0 mod 1 i 1 mod 0 i n n +1, e f (1) = n 1 Therefore, the graph, C n, is a 1-near mean For example, the 1-near mean cordial graph of C 5, is shown in the figure (b) For example,the 1-near mean cordial graph of C 8 is shown in the figure (a) Case(iii): when n 3 mod 4 f(u i 1 ) = n i, 1 i n 1 f(u i ) = i, 0 i n 1 f*(u n 1 u 0 ) = 1 0 i 0 mod 1 i 1 mod 0 i n Case(ii): when n 1 mod 4 f(u i 1 ) = n i, 1 i n 1 f(u i ) = i, 0 i n 1 Here, e f (0) = n 1, e f (1) = n+1 Therefore, the graph, C n, is a 1-near mean ISSN: All Rights Reserved 016 IJSETR 655

4 For example, the 1-near mean cordial graph of C 7 is shown in the figure (c), 0 i 1 mod f*( u i v j )= 1 i 0 mod for 0 i m 1, 0 j n 1 Here, e f (0)= e f (1)= mn mn +1 mn mn 1 Therefore, the graph, K m,n, is a 1-near mean Theorem 3.3. The complete bipartite graph, K m,n is a 1-Near Let G be K m,n. Let V(G) = {u i : 0 i m 1, v j : 0 j n 1} and E(G)={(u i v j : 0 i m 1, 0 j n 1} f(u i ) = i, 0 i m 1 f(v j ) = m + j, 0 j n 1 Case(i): when m is even and or odd 0 j 1 mod f*( u i-1 v j )= 1 j 0 mod for 1 i m, 0 j n 1 Illustration. The 1-near mean cordial graph of K 4,3 and K 3,3 are shown in the figure 3(a) and figure 3(b) 0 j 0 mod f*( u i v j )= 1 j 1 mod for 0 i m 1, 0 j n 1 Here, e f (0) = e f (1) = mn Case(ii): when m is odd and or odd 0 j 0 mod f*( u i-1 v j )= 1 j 1 mod for 1 i m 1, 0 j n 1 ISSN: All Rights Reserved 016 IJSETR 656

5 Theorem 3.4. The star K 1,n is a 1-Near Proof. Let G=(V, E) be a simple graph. Let G be K 1,n Let V(G)= {u, v i : 1 i n} and E(G) = {uv i : 1 i n}. Define f:v (G) {0, 1,..,.p 1}by f(u) = 0 f(v i ) = i, 1 i n, 0 i 0 mod f*( uv i )= 1 i 1 mod 1 i n n Here, e f (0)= n 1 e f (1)= n n +1 e f (0) e f (1) 1 Therefore, a star K 1,n is a 1-near mean Illustration 3. The 1-near mean cordial graph of K 1,5 is shown in the figure 4. f(u i+1 ) = i, 0 i n 1, f(u i ) = n i, 1 i n 1, f(v) = n or p 1. 0 i 1 mod 1 i 0 mod 1 i n 1 0 i 0 mod 3 f*( u i v)= 1 i 1, mod 3 1 i n Here, e f (0) = n 1, e f (1) = n Case(ii) when f(u i+1 )= i, 0 i n 1, f(u i ) = n i, 1 I n, f(v) = n or p 1. 0 i 0 mod 1 i 1 mod 1 i n 1 0 i 0,1 mod 4 f*( u i v)= 1 i,3 mod 4 1 i n Here, e f (0) = n 1, e f (1) = n Therefore, the graph P n + K 1 is a 1-near mean Illustration 4. The 1-near mean cordial graph of P 5 + K 1 and P 6 + K 1 are shown in the figure 5(a) and 5(b) Theorem 3.5. The graph P n +K 1 is a 1-Near Let G be P n + K 1 Let V (G) = {u i : 1 i n, v} and E(G)={[(u i u i+1 ) : 1 i n 1] [(u i v) : 1 i n]} Case:(i) when ISSN: All Rights Reserved 016 IJSETR 657

6 Theorem 3.6. The graph C n ʘK 1 is a 1-Near Let G be C n ʘK 1 Let V (G) = {u i : 0 i n, v i : 1 i n} and E(G)={[(u i u i+1 ) : 1 i n 1] (u n u1) [(u i v i ) : 1 i n]} f(u i ) = i, 1 i n f(v i ) = i 1, 1 i n f*(u i u i+1 ) = 0, 1 i n 1 f*(u n u 1 ) = 0 f*(u i v i ) = 1, 1 i n Here, e f (0) = e f (1) = n Therefore, the graph, C n ʘK 1 is a 1-near mean Let V (G) = {u i : 1 i n, v i : 1 i n} and E(G)={[(u i u i+1 ) : 1 i n 1] [(u i v i ) : 1 i n]} f(u i ) = i, 1 i n f(v i ) = i 1, 1 i n f*(u i u i+1 ) = 0, 1 i n 1 f*(u i v i ) = 1, 1 i n Here, e f (0) = n 1, e f (1) = n. Therefore, the graph, P n ʘ K 1, is a 1-near mean Illustration 6. The 1-near mean cordial graph of P 6 ʘ K 1 is shown in the figure 7 Illustration 5. The 1-near mean cordial graph of C 7 ʘK 1 is shown in the figure 6 Theorem 3.7. The graph P n ʘ K 1 is a 1-Near Let G be P n ʘ K 1 Theorem 3.8. The graph D (P n ) is a 1-Near Let G be D (P n ) Let V(G)={u i : 0 i n 1, v i : 0 i n 1} and E(G)={[(u i u i+1 ),(v i v i+1 ) : 0 i n ] [(u i v i+1 ), (v i u i+1 ) : 0 i n ]} f(u i ) = i, 0 i n 1 f(v i ) = i + 1, 0 i n 1 f*(u i u i+1 ) = 0, 0 i n f*(v i v i+1 ) = 0, 0 i n f*(u i v i+1 ) = 1, 0 i n ISSN: All Rights Reserved 016 IJSETR 658

7 f*(v i u i+1 ) = 1, 0 i n Here, e f (0) = n = e f (1) Therefore, the graph, D (P n ), is a 1-near mean Illustration 7. The 1-near mean cordial graph of D (P 3 ) is shown in the figure 8 [5] Nellai Murugan.A and Esther.G, Some Results on Mean Cordial Graphs, International Journal of Mathematics Trends and Technology, ISSN: , Volume 11, No., July 014 [6] Palani.K, Rejila Jeya Surya.J, 1-Near Mean Cordial Labeling of Graphs, IJMA- 6(7), Jully 015 PP15-0 [7] Raja Ponraj, Muthirulan Sivakumar and Murugesan Sundaram, Mean Cordial Labeling Of Graphs, Open Journal of Discrete Mathematics, 01,, References [1] Albert Williami, Indra Rajasingh and Roy.S, Mean Cordial Labeling of Certain Graphs, Journal of Computer and Mathematical Sciences, vol 4, Issue 4, 31 Augest, 013 pages(01-31) [] Gallian.J.A, A Dynamic Survey of Graph Labeling. The Electronic Journal of Combinatorics 6, #D4,5S6, 001. [3] Harary.F, Graph Theory, Addison- Wesley Publishing Company Inc,USA, 1969 [4] Nellai Murugan.A and Esther.G, Path related Mean Cordial Graphs, Journal of Global Research in Mathematical Archives, ISSN:30-58, Volume 11, No.3, March 014 ISSN: All Rights Reserved 016 IJSETR 659