CHAPTER 6 ODD MEAN AND EVEN MEAN LABELING OF GRAPHS

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1 92 CHAPTER 6 ODD MEAN AND EVEN MEAN LABELING OF GRAPHS In this chapter we introduce even and odd mean labeling,prime labeling,strongly Multiplicative labeling and Strongly * labeling and related results are presented. Further it is proved that comb, odd cycle, star and path graphs are odd and mean graphs. One vertex union of t isomorphic and non isomorphic cycles, cycle with pendent vertices, wheel graph when n is odd, are prime graphs. prism graph is Strongly Multiplicative and Strongly * graph. 6.1 INTRODUCTION A function f is called an Even Mean Labeling of a graph G with p vertices and q edges. If f is an injection from the vertices of G to the set {2,4,6,.2q} such that when each edge uv is assigned the label [f(u)+f(v)/2],then the resulting edge labels are distinct. A graph, which admits an Even Mean labeling, is said to be Even Mean Graph. A function f is called an Odd mean labeling of a graph G with p vertices and q edges. If f is an injection from the vertices of G to the set

2 93 {1,3,5,..2q-1} such that when each edge uv is assigned the label [f(u)+f(v)/2], then the resulting edge labels are distinct. A graph, which admits an odd mean labeling, is said to be odd mean graph. A graph with p vertices is Strongly Multiplicative if the vertices of G can be labeled with distinct integers 1,2,3,..p such that the labels induced on the edges by the product of the end vertices are distinct. A graph of order n is said to be a Strongly * graph if its vertices can be assigned the values 1,2,3,4..n in such a way that, when an edge whose vertices are labeled i and j is labeled with the value i+j+ij, all edges have different labels. 6.2 EVEN AND ODD MEAN GRAPH In this section, it is proved that comb P n Θ K 1, n C 2n+1, n K 1,n, n and P n, n are even and odd mean graphs. Theorem 6.1 Every Comb graph is odd and even mean graph. Proof: Let G = P n Θ K 1, n be a comb graph with 2n vertices and 2n-1 edges. The graph G is shown in the following Figure 6.1.

3 94 Figure 6.1 The comb graph P n Θ K 1 The even mean labeling for vertices of comb P n Θ K 1 is defined by u i = 4i 2, i = 1,2, n v i = 4i, i = 1,2, n Edge labelings are defined by e i = 4i, i = 1,2, n 1 e i = 4i 1, i = 1,2, n

4 95 The odd Mean labeling for vertices of comb P n Θ K 1 is defined by u i = 4i 3, i = 1,2, n v i = 4i 1, i = 1,2, n Edge labelings are defined by e i = 4i 1, i = 1,2, n 1 e i = 4i 2, i = 1,2, n From above, the labeling of vertices and edges are distinct. Hence the graph is even and odd mean graph. Example 6.1 Figure 6.2 The comb Graph P 9 Θ K 1

5 96 Example 6.2 Figure 6.3 The comb Graph P 12 Θ K 1 Theorem 6.2: Every cycle of odd length is even and odd mean graph Proof: Let G = C n, n 3 and n is odd The even mean labeling for vertices of C n is defined by u i = 2i, i =1,2,.n

6 97 Edge labelings are defined by e i = 2i +1, i = 1,2,.n-1, e n = n+1. The odd mean labeling for vertices of cycle C n is defined by u i = 2i-1,i = 1, 2, 3, 4..n Edge labelings are defined by e i = 2i, i = 1,2,3.n-1. e n = n. From the above assignment, the labeling of vertices and edges are distinct. Hence the graph is even and odd mean graph. Figure 6.4 The Graph Cn

7 98 Example 6.3 Every cycle of odd length is an even mean graph Figure 6.5 The Graph C9

8 99 Example 6.4 Every cycle of odd length is an odd mean graph Figure 6.6 The Graph C9 From above, the labeling of vertices and edges are distinct. Hence the graph is even and odd mean graph. Theorem 6.3: Every star graph is odd and even mean graph edges. Proof: Let G = K 1,n n be a star graph with n + 1 vertices and n

9 100 The odd mean la1beling for vertices of star graph K 1,n is defined by v 0 = 1, u i = 2i+1, i = 1,2,3..n Edge labelings are defined by e i = i + 1, i = 1,2,3 n. defined by The even mean labeling for vertices of star graph K 1,n, n is v 0 = 2, u i = 2i+2, i = 1,2,3.n Edge labeling are defined by e i = i +2, i = 1,2,3.n From the above assignment, the vertex and edge labeling are distinct. Hence the star graph is odd and even mean graph. Figure 6.7 The Star Graph

10 101 Example 6.5 Star graph K 1,12 is odd and even mean graph Figure 6.8 The Graph K 1, 12 Figure 6.9 The Graph K 1,12 graph. Theorem 6.4: Every path graph P n, n is odd and even mean edges. Proof: Let G = P n, n be a path graph with n vertices and n-1 The odd mean labeling for vertices of path graph P n is defined by u i = 2i-1, i = 1,2,3..n Edge labelings are defined by e i = 2i, i = 1,2, n - 1

11 102 defined by The even mean labeling for vertices of path graph P n n is u i =2i, i =1,2.n. e i = 2i+1, i =1,2,3 n - 1 From the above assignment, the vertex and edge labelings are distinct. Hence the graph G = P n, n is odd and even mean graph. Figure 6.10 The Graph P n

12 103 Example 6.6 Figure 6.11 The Graph P 7 Figure 6.12 The Graph P PRIME GRAPHS In this section, it is proved that the graph C n t (isomorphic), n n 3, t 1, C t n (non isomorphic), n n 3, t 1, W n where n is odd and C n.k 1, n are prime graphs.[16] Theorem 6.5: The one vertex union of t isomorphic copies of any cycle is prime graph Proof: Let G = C n t, n 2, t 1, a graph is one vertex union of t isomorphic copies of any cycle C n with tn+1 vertices and (n)t edges.

13 104 The prime labeling for vertices V ij. V 00 = 1,V i1 =i+1 for 1 i n V ij = n(j-1)+i+1 for 1 i n 2 j t The corresponding edges e 00 = gcd ( V 00, Vnj )=1 for 1 j t e ij = gcd (V ij,v i-1,j ) = 1 for 1 i n,1 j t From the above assignment, the vertices of G have prime labeling. Hence the graph G = C t n, n 3,t 1 is prime graph. Figure 6.13 The Graph C t n

14 105 Example 6.7 One vertex union of cycles of equal odd length is prime Figure 6.14 The Graph C 4 5

15 106 Example 6.8 One vertex union of cycles of equal even length is prime Figure 6.15 The Graph C 5 6 Theorem 6.6: The one vertex union of t non isomorphic cycles of different length is prime. Proof: Let C t n, n 2 and t 1, a graph is one vertex union of t non isomorphic copies of C n with [ (t+1)(t+2)/2] vertices and [{(t+3)(t+2)/2} 3] edges 1 i n, V 0,0 = 1 The prime labeling for vertices V ij = [i+j+(j-1)j/2], 1 j t, The corresponding edges e ij = gcd(v ij,v i-1,j ) =1, 1 i n and 1 j t.

16 107 From the above assignment, the vertices of G has prime labeling. Hence the one vertex union of t non isomorphic cycles of length 3,4,5. is prime graph. The graph is shown in the following Figure 6.16 Figure 6.16 The Graph C n t (non isomorphic)

17 108 Example 6.9 One vertex union of cycles of different length is prime Figure 6.17 The Graph C n t (non isomorphic) Theorem 6.7: Every cycle with pendent vertices is prime. Proof: Let G=C n.k 1, n be a cycle graph with pendent vertices has 2n vertices and 2n edges. The graph G is shown in the following Figure 6.18

18 109 Figure 6.18 The Graph C n.k 1 The prime labeling for vertices of cycle with pendent vertices C n.k 1 n is defined by u i = 2i-1, i = 1,2,3,, n v i = 2i, i = 1,2,3,, n Edge labelings are defined by e i = gcd(n,n+2) where n is odd e' = gcd(n, n+1) where n is odd From the above assignment, the vertices of G has prime labeling. Hence the graph G = C n.k 1, n n is prime.

19 110 Example 6.10 Figure 6.19 The Graph C 22.K 1 odd and n 4. Theorem 6.8: Every Wheel graph W n is prime graph when n is 2(n-1) edges. Proof: Let G = W n, n 4 be a wheel graph with n+1 vertices and The prime labeling for the vertices of wheel graph G is V 0 = 1. V i = i i n - 1. From the above assignment the vertex labelings are distinct. Consequently, we get edge labeling. The labelings are defined by e i י P = gcd (v, v ), e = gcd (v,v +1) e` = e =1 i 0 i i i i i

20 111 From the above assignment, the vertices of graph G has prime labeling. Hence the graph G =W n where n is odd is prime graph. The graph G is shown in Figure Figure 6.20 The Graph W n

21 112 Example 6.11 Wheel Graph W n is prime when n is odd Figure 6.21 The Graph W 13 graph. Theorem 6.9: The graph C n t (C n ) for n 3, t 3 is prime Proof:-Let G = C n t (C n ), n 3, t 3 be a graph with n(t-1) vertices and nt edges. The graph C n t (C n ) is shown in the following Figure 6.16 u 0 =1 The prime labeling for vertices of the graph C n t (C n ) is defined by u i = i + 1 for 1 i n(t-1) + 1, 1 j n

22 113 The corresponding edge labels are e i = gcd (u i,u i+1 ) = 1, 1 i n(t-1) + 1, 1 j n. From the above assignment, the vertices of G has prime labeling. Hence the graph G = C n t (C n ) n 3, t 3. The graph is shown in Figure 6.22 Figure 6.22 The Graph G = C n t (C n )

23 114 Example 6.12 Figure 6.23 The Graph C (C 10 )

24 115 Example 6.13 Figure 6.24

25 116 Example 6.14 Figure 6.25

26 117 One edge union of t non isomorphic cycles of even length is prime graph. Example 6.15 Figure STRONGLY MULTIPLICATIVE GRAPHS In this section results on Strongly Multiplicative labeling and Strongly *labeling are proved. Theorem 6.9: Every prism graph n(n+1)/2 n is strongly multiplicative. 3n(n+1)/2 edges. Proof: Let G be a prism graph with n(n+1)/2 vertices and

27 118 The graph G is shown in the following Figure 6.27 Figure 6.27 The Prism Graph n(n+1)/2 n is defined by The strongly multiplicative labeling for vertices of Prism n(n+1)/2 u i = i, i = 1,2. n(n+1)/2 Edge labelings are defined by e ij = u i u j distinct. From the above assignment, the labeling of vertices and edges are Hence the prism graph is strongly multiplicative.

28 119 Example 6.16 Figure 6.28 The Prism Graph 15 Theorem 6.10: Every prism graph n(n+1)/2 n is strongly * graph 3n(n+1)/2 edges. Proof: Let G be a prism graph with n(n+1)/2 vertices and defined by The strongly *- graph labeling for vertices of Prism n(n+1)/2 is u i = i, i = 1,2. n(n+1)/2

29 120 Edge labelings are defined by e ij = u i + u j +u i u j From the above assignment, the labeling of vertices and edges are distinct. Hence the prism graph is strongly * multiplicative. Example 6.17 Figure 6.29 The Prism Graph 15

30 CONCLUSION In this chapter, it is proved that Comb P n K 1 n, Cycle C 2n+1 n, Star K 1,n n are even and odd mean graphs. One vertex union of t isomorphic and non isomorphic cycles, cycle with pendent vertices, Wheel graph W n when n is odd are prime graphs. Prism graph is strongly Multiplicative graph and strongly * graph.