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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 Θ 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 Θ 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 Θ K1

The even mean labeling for vertices of comb Pn Θ 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 Θ 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 Θ K1

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Example 6.2

Figure 6.3 The comb Graph P12 Θ 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 = 1,2,3…n.

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 =1,2,3……n - 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 , ∀ n≥2 and t≥1, 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, 1≤i≤n and 1≤j≤t.

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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

eiPי = 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, t≥3 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 prime labeling for vertices of the graph Cn t (Cn ) is defined by

u0 =1

ui = i + 1 for 1 ≤ i ≤ n(t-1) + 1, 1 ≤ j ≤n

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The corresponding edge labels are

ei = gcd (ui ,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 = Cn t (Cn) ∀ n≥3, t≥3.

The graph is shown in Figure 6.22

Figure 6.22 The Graph G = Cn t (Cn)

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Example 6.12

Figure 6.23 The Graph C1010 (C10)

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Example 6.13

Figure 6.24

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Example 6.14

Figure 6.25

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One edge union of ‘t’ non isomorphic cycles of even length is prime

graph.

Example 6.15

Figure 6.26

6.4 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.

Proof: Let G be a prism graph with n(n+1)/2 vertices and

3n(n+1)/2 edges.

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The graph G is shown in the following Figure 6.27

Figure 6.27 The Prism Graph ∆ n(n+1)/2

The strongly multiplicative labeling for vertices of Prism ∆ n(n+1)/2

n is defined by ∀

ui = i, i = 1,2 …. n(n+1)/2

Edge labelings are defined by eij = ui uj

From the above assignment, the labeling of vertices and edges are

distinct.

Hence the prism graph is strongly multiplicative.

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Example 6.16

Figure 6.28 The Prism Graph ∆ 15

Theorem 6.10: Every prism graph ∆ n(n+1)/2 ∀ n is strongly

* graph

Proof: Let G be a prism graph with n(n+1)/2 vertices and

3n(n+1)/2 edges.

The strongly *- graph labeling for vertices of Prism ∆ n(n+1)/2 is

defined by

ui = i, i = 1,2 …. n(n+1)/2

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Edge labelings are defined by

eij = ui + uj+ui uj

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

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6.5 CONCLUSION

In this chapter, it is proved that Comb Pn K1 ∀ 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 Wn when n is odd are prime graphs. Prism graph is strongly

Multiplicative graph and strongly * graph.