One Modulo Three Harmonic Mean Labeling of Graphs

International Journal of Mathematics Research. ISSN 0976-5840 Volume 5, Number 4 (20), pp. 4-422 International Research Publication House http://www.irphouse.com One Modulo Three Harmonic Mean Labeling of Graphs C. David Raj 2 S.S. Sandhya and C. Jayasekaran Department of Mathematics, Malankara Catholic College, Mariagiri, Kanyakumari District. Email: davidrajmccm@gmail.com Department of Mathematics, Sree Ayyappa College for Women Chunkankadai; Kanyakumari-629807, India. Email: sssandhya2009@gmail.com Department of Mathematics, Pioneer Kumaraswamy College, Nagercoil, Kanayakumari 62900, TamilNadu, India. Email: jaya_pkc@yahoo.com Abstract In this paper, we introduce a new labeling called one modulo three harmonic mean labeling. A graph G is said to be one modulo three harmonic mean graph if there is a function φ from the vertex set of G to {,, 4,,q 2, q} with φ is one-one and φ induces a bijection φ from the edge set of G to {, 4,, q - 2}, where φ (e = uv) = ( ) ( ) or ( ) ( ) and the function φ is ( ) ( ) ( ) ( ) called as one modulo three harmonic mean labeling of G. Further, we prove that some graphs are one modulo three mean graphs Key words: One modulo three harmonic mean labeling, one modulo three harmonic mean graphs. Introduction and definitions All graphs considered here are simple, finite, connected and undirected. The symbols V(G) and E(G) denote the vertex set and the edge set of a graph G. A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions. Several types of graph labeling and a detailed survey is available in []. For standard terminology and notations we follow harary[2 ]. V.swaminathan and C. Sekar introduce the concept of one modulo three graceful labeling in []. S.Somasundaram and R.Ponraj introduced mean labeling of graphs in

42 C. David Raj et al [4]. S.Somasundaram and S.S Sandhya introduced the concept of harmonic mean labeling in [5] and studied their behavior in [6], [7] and [8]. In this paper,we introduce a new type of labeling known as one modulo three harmonic mean labeling and investigate one modulo three harmonic mean graphs. We will provide brief summary of definitions and other information which are necessary for the present investigation. Definition.: The corona G G 2 of two graphs G and G 2 is defined as the graph G obtained by taking one copy of G (which has p vertices) and p copies of G 2 and then joining the i th vertex of G to every vertices in the i th copy of G 2. Definition.2: The graph P n K is called Comb. Definition.: The graph C n K is called crown. 2. One modulo three harmonic mean labeling Definition 2.: A graph G is said to be one modulo three harmonic mean graph if there is a function φ from the vertex set of G to {,, 4, 6,,q 2, q} with φ is one-one and φ induces a bijection φ from the edge set of G to {, 4,, q - 2}, where φ (e = uv) = ( ) ( ) or ( ) ( ) and the function φ is called as one ( ) ( ) ( ) ( ) modulo three harmonic mean labeling of G. Remark 2.2: If G is a one modulo three harmonic mean graph, then must be a label of one of the vertices of G, since an edge must get the label. Theorem 2.: If G is a one modulo three harmonic mean graph, then there is atleast one vertex of degree one. Suppose G is a one modulo three harmonic mean graph. Then by remark 2.2, must be the label of one of the vertices of G. If the vertex u get the label, then any edge incident with u get the label or 2, since < m. But 2 is not in the set {,, 4,,q 2, q}. Therefore, this vertex must have degree one. Hence the theorem. Theorem 2.4: Any k-regular graph, k>, is not one modulo three harmonic mean graph. Let G be a k-regular graph. Suppose G is one modulo three harmonic mean graph. Then by theorem 2., there is atleast one vertex of degree one. Which is a contradiction to k>. Hence the theorem.

One Modulo Three Harmonic Mean Labeling of Graphs 4 Corollary 2.5: Any cycle C n is not a one modulo three harmonic mean graph. The proof follows from the theorem 2.4. Theorem 2.6: Any path P n is a one modulo three harmonic mean graph. Let P n be the path u u 2 u n. Define a function φ: V(G) {,, 4, 6,, q 2, q} by φ(u ) =, φ(u i ) = (i ), 2 i n. Then φ induces a bijection φ : E(G) {, 4,, q - 2},where φ (u i u i+ ) = i 2, i n. Therefore, φ is a one modulo three harmonic mean labeling. Hence P n is a one modulo three harmonic mean graph. Example 2.7: One modulo three harmonic mean labeling of P 8 is shown in figure. u u 2 4 u 6 7 u 4 9 0 u 5 2 u 6 5 6 u 8 9 u 7 8 2 Figure Theorem 2.8:A comb P n K is a one modulo three harmonic mean graph. Let u, u 2,, u n, v, v 2,, v n be the vertices of P n K. Here we consider two cases. Case (): n is odd. Define a function f: V(P n K ) {,, 4, 6,, q -2, q} by φ(u ) = φ(u i ) = 6i, for all i = 2, 4,, n φ(u i ) = 6(i ), for all i =, 5,, n φ(v ) = φ(v i ) = 6(i ), for all i = 2, 4,, n φ(v i ) = 6i, for all i =, 5,, n. Then φ induces a bijective function φ : E(P n K ) {, 4, 7,, q 2},where φ (u i u i+ ) = 6i 2, i n φ (u i v i ) =6i 5, i n Thus the edges get the distinct labels, 4,, q 2. In this case φ is a one modulo three harmonic mean labeling.

44 C. David Raj et al Case (2): n is even. Define a function φ : V(P n K ) {,, 4, 6,, q -2, q} by φ(u ) = φ(u i ) = 6i, for all i = 2, 4,, n φ(u i ) = 6(i ), for all i =, 5,, n φ (u i ) = i, i n φ (v ) = φ (v i ) = 6(i ), for all i = 2, 4,, n φ (v i ) = 6i, for all i =, 5,, n. Then φ induces a bijective function φ : E(P n K ) {, 4, 7,, q 2},where φ (u i u i+ ) = 6i 2, i n φ (u i v i ) =6i 5, i n Thus the edges get the distinct labels, 4,, q 2. In this case φ is a one modulo three harmonic mean labeling. From case () and case (2) we conclude that P n K is a one modulo three harmonic mean graph. Example 2.9: One modulo three harmonic mean labeling of P 6 K and P 7 K is shown in figure 2 and figure respectively. 9 2 2 24 6 5 8 27 0 Figure 2 9 2 2 24 6 6 5 8 27 0 9 Figure We have P 2 = K 2, which is one modulo three harmonic mean graph. Next we have the following theorem. Theorem 2.0: If n > 2, K n is not a one modulo three harmonic mean graph.

One Modulo Three Harmonic Mean Labeling of Graphs 45 Suppose K n, n > 2 is a one modulo three harmonic mean graph. Then by theorem, there exists atleast one vertex of degree one. This is the contradiction to the fact that all the vertices are of degree greater than one. Hence the theorem. Theorem 2.: K,n is one modulo three harmonic mean graph if and only if n 6. K, is same as P 2 and K,2 is P. Hence by theorem 2. K, and K,2 are one modulo three harmonic mean graphs. One modulo three harmonic mean labeling for K,, K,4, K,5 and K,6 are displayed below. Figure 4 Assume n > 6 and suppose K,n has one modulo three harmonic mean labeling. Let φ(u) be the central vertex of K,n. Here we consider two cases. Case(): 2 φ(u). Then clearly there is no edge with label q 2,q > 6, since the highest edge label is... Hence in this case K,n is not a one modulo three harmonic mean graph. Case (2) :φ(u) 5. Then there is no edge with label 4 as shown in the following figure 5.

46 C. David Raj et al 5 5.. 6 or 7 4 Figure 5 Hence K,n is not a one modulo three harmonic mean graph for n > 6. Theorem 2.2: P n K 2 is a one modulo three Harmonic mean graph. Let u i, u i j, i n, j 2 be the vertices of P n K 2. Define a function φ: V(P n K 2 ) {,, 4, 6,, q 2, q} by φ(u ) = 4 φ(u i ) = 9i 5, 2 i n φ(u ) = φ(u i ) = 9(i ), 2 i n φ(u i 2 ) = 9i 6, i n. Then φ induces a bijective function φ : E(P n K 2 ) {, 4,, q 2}, where φ (u i u i+ ) = 9i 2, i n φ (u i u i ) = 9i 8, i n φ (u i u i 2 ) = 9i 5, i n. Therefore, φ is a one modulo three harmonic mean labeling of P n K 2. Hence P n K 2 is one modulo three harmonic mean graph. Example 2.: One modulo three harmonic mean labeling of P 5 K 2 is shown in figure 6. 9 2 8 2 27 0 6 9 4 22 40 Figure 6

One Modulo Three Harmonic Mean Labeling of Graphs 47 Theorem 2.4: A graph obtained by attaching a path of length two at each vertex of P n is one modulo three harmonic mean graph. Let u i be the vertices of P n and v i, w i, be the vertices path of length two, i n. Let G be a graph obtained by attaching a path of length two at each vertex of P n. Define a function φ: V(G) {,, 4, 6,, q 2, q} by φ(u i ) = 9i 5, i 2 φ(u i ) = 9i 6, i n φ(v i ) = 9i 6, i 2 φ(v i ) = 9i 5, i n φ(w ) = φ(w 2 ) = 9 φ(w i ) = 9(i 2) 2, i n. Then φ induces a bijective function φ : E(G) {, 4,, q 2}, where φ (u i u i+ ) = 9i 2, i n φ (u i v i ) = 9i 5, i n φ (u i w i ) = 9i 8, i n. Therefore, φ is one modulo three harmonic mean labeling of G. Hence G is one modulo three harmonic mean graph. Example 2.5: One modulo three harmonic mean labeling of G, When n=7 is shown in figure 7. 9 6 25 4 4 52 2 22 40 49 58 4 2 0 9 48 57 Figure 7 Theorem 2.6: C n K is one modulo three Harmonic mean graph. Let u i, v i, i n, be the vertices of C n K. Define a function φ: V(C n K ) {,, 4, 6,, q 2, q} by

48 C. David Raj et al Case(): if n is odd, φ(u ) = 7 φ(u i ) = 6i, i =, 5,, n φ(u 2 ) =5, φ(u i ) = 6i, i = 4, 6,,n φ(v ) = φ(v ) = 4 φ(v i ) = 6i, i = 5, 7,, n φ(v i ) = 6i, i = 4, 6,, n Case(2): if n is even, φ(u ) =7, φ(u i ) = 6i, i =, 5,, n φ(u 2 ) = 5 φ(u i ) = 6i, i = 4, 6,, n φ(v ) = φ(v ) = 4 φ(v i ) = 6i, i = 5, 7,, n φ(v 2 ) = φ(v i ) = 6i, i = 4, 6,, n. Then φ induces a bijective function φ : E(C n K ) {, 4,, q 2}, where φ (u i u i+ ) = 6i + 4, i 2 φ (u i u i+ ) = 6i +, i n φ (u i v i ) = i 2, i φ (u i v i ) = 6i 2, 4 i n. Therefore, φ is one modulo three harmonic mean labeling of C n K. Hence C n K is one modulo three harmonic mean graph. Example 2.7: One modulo three harmonic mean labeling of C 7 K is given in figure 9.

One Modulo Three Harmonic Mean Labeling of Graphs 49 9 42 7 5 6 8 27 0 2 4 24 Figure 9 Theorem 2.8: C n K 2 is one modulo three Harmonic mean graph. Let u i, u i j, i n, j 2 be the vertices of C n K 2. Define a function φ: V(C n K 2 ) {,, 4, 6,, q 2, q} by φ(u ) = 7 φ(u i ) = 9i, 2 i n φ(u ) = φ(u 2 ) = 4 φ(u i ) = 9i 6, i n φ(u 2 ) = φ(u i 2 ) = 9i, 2 i n. Then φ induces a bijective function φ : E(C n K 2 ) {, 4,, q 2}, where φ (u i u i+ ) = 9i +, i n φ (u u ) =, φ (u 2 u 2 ) =7, i n φ (u i u i ) = 9i 5, i n φ (u u 2 ) = 4 φ (u i u i 2 ) = 9i 2, 2 i n. Therefore, φ is one modulo three harmonic mean labeling of C n K 2. Hence C n K 2 is one modulo three harmonic mean graph.

420 C. David Raj et al Example 2.9: One modulo three harmonic mean labeling of C 5 K 2 is shown in figure 0. 54 7 4 48 5 5 8 45 42 24 2 9 27 6 0 Figure 0 Theorem 2.20: A graph obtained by attaching a path of length two at each vertex of C n admits One Modulo three Harmonic mean labeling. Let G be the graph obtained by attaching a path of length two at each vertex of C n. Let u i, v i, w i, i n be the vertices of G. Define a function φ: V(G) {,, 4, 6,, q 2, q} by φ(u ) = 7 φ(u i ) = 9i, 2 i n φ(v ) = 4 φ(v i ) = 9i 2, 2 i n φ(w ) = φ(w 2 ) = 4 φ(w i ) = 9i 8, i n. Then φ induces a bijective function φ : E(G) {, 4,, q 2}, where φ (u i u i+ ) = 9i +, i n φ (u v ) = 4 φ (u i v i ) = 9i 2, 2 i n φ (u w ) = φ (u 2 w 2 ) = 7 φ (u i w i ) = 9i 5, i n.

One Modulo Three Harmonic Mean Labeling of Graphs 42 Therefore, φ is one modulo three harmonic mean labeling of G. Hence G is one modulo three harmonic mean graph. Example 2.2: One modulo three harmonic mean labeling of G, n=6 is shown in figure. 46 52 5 7 5 6 4 7 4 42 24 25 4 9 28 Figure References: [] J.A Gallian, A dynamic survey of graph labeling, The electronic journal of combinatorics DS6, 202. [2] F.Harary,Graph theory, Addison Wesley, Massachusetts. [] V.Swaminathan and C.Sekar, Modulo three graceful graphs, proceed. National conference on mathematical and computational models, PSG College of Technology, Coimbatore, 200, 28 286. [4] S.Somasundaram, and R.Ponraj, Mean labeling of graphs, National Academy Science letters vol.26, (200), 20 2. [5] S.Somasundaram, S.S.Sandhya and R.Ponraj,Harmonic mean labeling of graphs, communicated to Journal of combinatorial mathematics and combinatorial computing. [6] S.S Sandhya, S.Somasundaram and R.Ponraj, Some more Results on Harmonic mean graphs, Journal of Mathematical Research 4() (202). [7] S.S Sandhya, S.Somasundaram and R. Ponraj, Some results on Harmonic mean Graphs, International Journal of Contemporary Mathematical Sciences (74) (202),97 208.

422 C. David Raj et al [8] S.S Sandhya, S.Somasundaram and R. Ponraj, Jarmonic mean labeling of some Cycle Related Graphs, International Journal of Mathematical Analysis,vol.6,202 No.7 40,997 2005.