CHAPTER 6 ODD MEAN AND EVEN MEAN LABELING OF GRAPHS

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    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

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    {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 Pn K1, n C 2n+1, n K

    1,n, n and Pn, n are even and odd mean graphs.

    Theorem 6.1 Every Comb graph is odd and even mean graph.

    Proof: Let G = Pn K1, n be a comb graph with 2n vertices and

    2n-1 edges. The graph G is shown in the following Figure 6.1.

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    Figure 6.1 The comb graph Pn K1

    The even mean labeling for vertices of comb Pn K1 is defined by

    ui = 4i 2, i = 1,2, n

    vi = 4i, i = 1,2, n

    Edge labelings are defined by

    ei = 4i, i = 1,2, n 1

    ei = 4i 1, i = 1,2, n

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    The odd Mean labeling for vertices of comb Pn K1 is defined by

    ui = 4i 3, i = 1,2, n

    vi = 4i 1, i = 1,2, n

    Edge labelings are defined by

    ei = 4i 1, i = 1,2, n 1

    ei = 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 P9 K1

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    Example 6.2

    Figure 6.3 The comb Graph P12 K1

    Theorem 6.2:

    Every cycle of odd length is even and odd mean graph

    Proof: Let G = Cn , n 3 and n is odd

    The even mean labeling for vertices of Cn is defined by

    ui = 2i , i =1,2, .n

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    Edge labelings are defined by

    ei = 2i +1, i = 1,2, .n-1, en = n+1.

    The odd mean labeling for vertices of cycle Cn is defined by

    ui = 2i-1,i = 1, 2, 3, 4..n

    Edge labelings are defined by

    ei = 2i, i = 1,2,3.n-1.

    en = 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 nC

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    Example 6.3

    Every cycle of odd length is an even mean graph

    Figure 6.5 The Graph 9C

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    Example 6.4

    Every cycle of odd length is an odd mean graph

    Figure 6.6 The Graph 9C

    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

    Proof: Let G = K 1,n n be a star graph with n + 1 vertices and n

    edges.

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    The odd mean la1beling for vertices of star graph K 1,n is defined by

    v0 = 1,

    ui = 2i+1, i = 1,2,3..n

    Edge labelings are defined by ei = i + 1, i = 1,2,3n.

    The even mean labeling for vertices of star graph K1,n , n is

    defined by

    v0 = 2, ui = 2i+2, i = 1,2,3.n

    Edge labeling are defined by

    ei = 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

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    Example 6.5

    Star graph K1,12 is odd and even mean graph

    Figure 6.8 The Graph K 1, 12 Figure 6.9 The Graph K 1,12

    Theorem 6.4: Every path graph Pn , n is odd and even mean

    graph.

    Proof: Let G = Pn , n be a path graph with n vertices and n-1

    edges.

    The odd mean labeling for vertices of path graph Pn is defined by

    ui = 2i-1, i = 1,2,3..n

    Edge labelings are defined by

    ei = 2i, i = 1,2,n - 1

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    The even mean labeling for vertices of path graph Pn n is

    defined by

    ui =2i, i =1,2.n.

    ei = 2i+1, i =1,2,3n - 1

    From the above assignment, the vertex and edge labelings are

    distinct. Hence the graph G = Pn , n is odd and even mean graph.

    Figure 6.10 The Graph Pn

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    Example 6.6

    Figure 6.11 The Graph P7 Figure 6.12 The Graph P7

    6.3 PRIME GRAPHS

    In this section, it is proved that the graph Cnt (isomorphic) ,

    n n 3, t 1, C tn (non isomorphic), n n 3, t 1, Wn where n is odd

    and Cn.K1 , 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 = Cnt , n 2, t 1, a graph is one vertex union of

    t isomorphic copies of any cycle Cn with tn+1 vertices and (n)t edges.

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    The prime labeling for vertices Vij. V00= 1,Vi1=i+1 for 1 i n

    Vij = n(j-1)+i+1 for 1 i n 2 j t

    The corresponding edges e00 = gcd ( V00, Vnj )=1 for 1 j t

    eij = gcd (Vij,Vi-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 = Ctn , n 3,t 1 is prime graph.

    Figure 6.13 The Graph Ctn

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    Example 6.7

    One vertex union of cycles of equal odd length is prime

    Figure 6.14 The Graph C45

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    Example 6.8

    One vertex union of cycles of equal even length is prime

    Figure 6.15 The Graph C56

    Theorem 6.6: The one vertex union of t non isomorphic cycles

    of different length is prime.

    Proof: Let Ctn , n2 and t1, a graph is one vertex union of t

    non isomorphic copies of Cn with [ (t+1)(t+2)/2] vertices and

    [{(t+3)(t+2)/2} 3] edges

    The prime labeling for vertices Vij = [i+j+(j-1)j/2], 1 j t,

    1 i n, V0,0 = 1

    The corresponding edges eij = gcd(vij,v i-1,j) =1, 1in and 1jt.

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    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 Cnt (non isomorphic)

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    Example 6.9

    One vertex union of cycles of different length is prime

    Figure 6.17 The Graph Cnt (non isomorphic)

    Theorem 6.7: Every cycle with pendent vertices is prime.

    Proof: Let G=Cn .K1 , 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

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    Figure 6.18 The Graph Cn.K1

    The prime labeling for vertices of cycle with pendent vertices Cn.K1

    n is defined by

    ui = 2i-1, i = 1,2,3,, n

    vi = 2i , i = 1,2,3,, n

    Edge labelings are defined by ei = 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 = Cn.K1, n n is prime.

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    Example 6.10

    Figure 6.19 The Graph C22 .K1

    Theorem 6.8: Every Wheel graph Wn is prime graph when n is

    odd and n 4.

    Proof: Let G = Wn , n 4 be a wheel graph with n+1 vertices and

    2(n-1) edges.

    The prime labeling for the vertices of wheel graph G is V0 = 1.

    Vi = i + 1 1 i n - 1.

    From the above assignment the vertex labelings are distinct.

    Consequently, we get edge labeling. The labelings are defined by

    ei P = gcd (v , v ), e = gcd (v ,v +1) e` = e =1 i 0 i i i i i

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    From the above assignment, the vertices of graph G has prime

    labeling.

    Hence the graph G =Wn where n is odd is prime graph.

    The graph G is shown in Figure 6.20.

    Figure 6.20 The Graph Wn

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    Example 6.11

    Wheel Graph Wn is prime when n is odd

    Figure 6.21 The Graph W13

    Theorem 6.9: The graph Cnt (Cn ) for n 3, t3 is prime

    graph.

    Proof:-Let G = Cn t (Cn ), n 3, t 3 be a graph with n(t-1)

    vertices and nt edges.

    The graph Cn t (Cn ) is shown in the following Figure 6.16

    The