International journal of scientific and technical research in engineering (IJSTRE) www.ijstre.com Volume 1 Issue 7 ǁ October 2016. Some New Prime Graphs

S.MEENA, J.NAVEEN Associate Professor of Mathematics Government Arts College, C.Mutlur, Chidambaram, Tamil Nadu, India E-mail : [email protected]

Assistant Professor of Mathematics Government Arts College, C.Mutlur, Chidambaram, Tamil Nadu, India E-mail : [email protected]

ABSTRACT : A Graph G with n vertices is said to admit prime labeling if its vertices can be labeled with distinct positive integers not exceeding n such that the labels of each pair of adjacent vertices are relatively prime. A graph G which admits prime labeling is called a prime graph. In this paper we investigate the ∗ existence of prime labeling of some graphs related to cycle Cn, wheel 푊푛 ,crown 퐶푛 ,Helm퐻푛and Gear graph퐺푛 ,Star푠푛 , 푇푛 ,prism 퐷푛 and Butterfly graph 퐵푛,푚 .We discuss prime labeling in the context of the graph operation namely duplication.

Keywords:Graph Labeling, Prime Labeling, Duplication, Prime Graph.

I. INTRODUCTION We begin with simple, finite, connected and undirected graph G (V,E) with p vertices and q edges.The set of vertices adjacent to a u of G is denoted by N(u).For notations and terminology we refer to Bondy and Murthy[1]. The notion of prime labeling was introduced by Roger Entringer and was discussed in a paper by Tout[6]. Two integers a and b are said to be relatively prime if their greatest common divisor is 1. Relatively prime numbers play an important role in both analytic and algebraic number theory. Many researchers have studied prime graph.Fu. H [3] has proved that the path 푃푛 onn vertices is a prime graph. Deretsky el al [2] haveprove that the cycle 퐶푛 on n vertices is a prime graph. Around 1980 rugerEtringer conjectured that all trees have prime labeling which is not settled till today. The prime labeling for planer grid is investigated by Sundaram et al [5] , Lee.S.al [4] has proved that the Wheel 푊푛 is a prime graph if and only if n is even.

′ " Definition 1.1[7] Duplication of a vertex 푣푘 by a new edge e = 푣푘 푣푘 in a graph G produces a new graph G' such ′ " that N(푣푘 ) ∩ N(푣푘 )= 푣푘

Definition 1.2 The graph obtained by duplicating all the vertices by edges of a graph G is called duplication of G

∗ Definition 1.3 The crown graph 퐶푛 is obtained from a cycle Cn by attaching a pendent edge at each vertex of the n-cycle.

Definition 1.4 The Helm퐻푛 is a graph obtained from a Wheel by attaching a pendent edge at each vertex of the n-cycle.

Definition 1.5The gear graph퐺푛 is, the graph obtained from wheel 푊푛 = 퐶푛 + 퐾1by subdividing each edge incident with the apex vertex once.

Definition 1.6TheFriendship graph 푇푛 is set of n triangles having a common central vertex.

Definition 1.7The Prism 퐷푛 is a Graph obtained from the cycleCn(= 푣1,푣2 , … , 푣푛 )by attaching n-3 chords푣1푣3 , 푣1푣4,… , 푣1푣푛−2. Manuscript id. 249778 www.ijstre.com Page 9 Some New Prime Graphs

Definition 1.8 The Butterfly Graph 퐵푛,푚 is,the Graph obtained from two copies ofCn having one vertex in common, and by attaching m pendent edges tothe common vertex of two cycles.

In this paper we proved that the graphs obtained by duplication of all the alternate vertices by edges inCnand ∗ wheel 푊푛 if n is even , duplicatingall the rim vertices byedges in crown 퐶푛 ,Helm 퐻푛 andGear graph퐺푛 , duplicating all the alternate vertices by edges in star 푆푛 = 퐾1,푛 and Friendship graph 푇푛 and prism 퐷푛 ,duplicating all the vertices of 퐶푛 by edges in Butterfly graph 퐵푛,푚 , m ≥ 2n-2 are all prime graphs.

II. MAIN RESULTS Theorem 2.1 The graph obtained by duplicating all the alternate vertices by edges in퐶푛 is a prime graph , if n is even. Proof. Let푉(퐶푛 ) = {푢푖 / 1 ≤ 푖 ≤ 푛} 퐸(퐶푛 ) = {푢푖 푢푖+1 / 1 ≤ 푖 ≤ 푛} υ {푢푛 푢1 } Let G be the graph obtained by duplicatingall the alternate vertices by edges in퐶푛 and let the new edges ′ " ′ " ′ " be푢1푢1, 푢3푢3,… , 푢푛−1푢푛−1by duplicating the alternate vertices 푢1, 푢3 , … . 푢푛−1respectively, ′ " Then푉 퐺 = { 푢푖 / 1 ≤ 푖 ≤ 푛} ∪ 푢푖 , 푢푖 /1 ≤ 푖 ≤ 푛, 푖 푖푠 표푑푑 ′ " ′ " 퐸 퐺 = {푢푖 푢푖+1 /1 ≤ 푖 ≤ 푛 − 1} ∪ {푢푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖 / 1 ≤ 푖 ≤ 푛 , 푖 푖푠 표푑푑} ∪ { 푢푛 푢1} 푉 퐺 = 2푛, 퐸 퐺 = 5푛/2 Define a labeling 푓 ∶ 푉 퐺 → 1,2,3,… , 2푛 as follows. Let 푓 푢1 = 1 푓 푢푖 = 2푖 푖푓 푖 푖푠 푒푣푒푛 , 2 ≤ 푖 ≤ 푛, 푓 푢푖 = 2푖 − 1 푖푓 푖 푖푠 표푑푑 , 3 ≤ 푖 ≤ 푛 − 1 푎푛푑 푖 ≢ 2 푚표푑 3 " 푓 푢푖 = 2푖 + 1 , 푖푓 푖 푖푠 표푑푑 , 1 ≤ 푖 ≤ 푛 − 1 푎푛푑 푖 ≡ 2 푚표푑 3 푓 푢푖 = 2푖 − 1 , 푖푓 푖 푖푠 표푑푑 , 1 ≤ 푖 ≤ 푛 − 1 푎푛푑 푖 ≡ 2 푚표푑 3 ′ 푓 푢푖 = 2푖, 푖푓 푖 푖푠 표푑푑 , 1 ≤ 푖 ≤ 푛 − 1 " 푓 푢푖 = 2푖 + 1 , 푖푓 푖 푖푠 표푑푑 , 1 ≤ 푖 ≤ 푛 − 1 푎푛푑 푖 ≢ 2 푚표푑 3 Since 푓 푢1 = 1 ′ gcd 푓(푢1), 푓 푢2 =1, gcd 푓 푢1 , 푓 푢1 =1, " gcd 푓 푢1 , 푓 푢1 = 1, gcd 푓(푢푛 ), 푓 푢1 = 1.

gcd 푓 푢푖 , 푓 푢푖+1 = gcd 2푖 , 2 푖 + 1 − 1 = gcd 2푖, 2푖 + 1 = 1, 푖 푖푠 푒푣푒푛 , 푓표푟 2 ≤ 푖 ≤ 푛 , 푖 ≢ 2 푚표푑 3 If i is odd .

gcd 푓 푢푖 , 푓 푢푖+1 = gcd 2푖 − 1 , 2 푖 + 1 = gcd 2푖 − 1 , 2푖 + 2 = 1 푓표푟 3 ≤ 푖 ≤ 푛 , 푖 ≢ 2 푚표푑 3 as among these two numbers one is odd and other is even and their difference is 3 and they are not multiples of 3

gcd 푓 푢푖−1 , 푓 푢푖 = gcd 2 푖 − 1 , 2푖 + 1 = gcd 2푖 − 2 , 2푖 + 1 = 1 푓표푟 3 ≤ 푖 ≤ 푛 , 푖 ≡ 2 푚표푑 3 as among these two numbers one is even and other is odd their difference is 3 . And they are not multiples of 3

gcd 푓 푢푖 , 푓 푢푖+1 = gcd 2푖 + 1 , 2푖 + 2 = 1 푓표푟 3 ≤ 푖 ≤ 푛 , 푖 ≡ 2 푚표푑 3 ′ gcd 푓푢푖 ), 푓 푢푖 = gcd 2푖 − 1, 2푖 = 1 푓표푟 3 ≤ 푖 ≤ 푛, 푖 ≡ 2 푚표푑 3 as these two number are consecutive integers " gcd 푓 푢푖 , 푓 푢푖 = gcd 2푖 − 1, 2푖 + 1 =1 푓표푟 3 ≤ 푖 ≤ 푛, 푖 ≢ 2 푚표푑 3 " gcd 푓 푢푖 , 푓 푢푖 = gcd 2푖 + 1, 2푖 − 1 = 1 푓표푟 3 ≤ 푖 ≤ n, 푖 ≡ 2 푚표푑 3 as these two number are odd consecutive integers ′ " gcd 푓 푢푖 ,푓 푢푖 = gcd 2푖, 2푖 + 1 = 1 푓표푟 1 ≤ 푖 ≤ n, 푖 ≢ 2 푚표푑 3 ′ " gcd 푓 푢푖 ,푓 푢푖 = gcd 2푖, 2푖 − 1 = 1 푓표푟 1 ≤ 푖 ≤ n, 푖 ≡ 2 푚표푑 3 as these two number are consecutive integers Thus푓 is a prime labeling. HenceG is a prime graph.

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Theorem 2.2 The graph obtained by duplicating all the alternate rim vertices by edges in Wheel 푊푛 is a prime graph. If n is even and 푛 ≢ 1 푚표푑 3 . Proof. Let푉(푊푛 ) = { 푐,푢푖 / 1 ≤ 푖 ≤ 푛} 퐸(푊푛 ) = { 푐푢푖 / 1 ≤ 푖 ≤ 푛 } υ {푢푖 푢푖+1 / 1 ≤ 푖 ≤ 푛 − 1 } υ { 푢푛 푢1 } Let G be the graph obtained by duplicating all the alternate rim vertices by edges in Wheel 푊푛 and let the new ′ " ′ " ′ " edges be 푢1푢1, 푢3푢3,… , 푢푛−1푢푛−1by duplicating the vertices 푢1, 푢3 , … . 푢푛−1respectively, ′ " 푉 퐺 = { 푐, 푢푖 / 1 ≤ 푖 ≤ 푛} ∪ 푢푖 , 푢푖 / 1 ≤ 푖 ≤ 푛 − 1, 푖 푖푠 표푑푑 ′ " ′ " 퐸 퐺 = 푐푢푖 /1 ≤ 푖 ≤ 푛 ∪ 푢푖 푢푖+1 /1 ≤ 푖 ≤ 푛 − 1 ∪ {푢푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖 /1 ≤ 푖 ≤ 푛, 푖푖푠표푑푑} ∪ { 푢푛 푢1} 푉 퐺 = 2푛 + 1, 퐸 퐺 = 7푛/2 Define a labeling 푓 ∶ 푉 퐺 → 1,2,3, … , 2푛 + 1 as follows Let 푓 푐 = 1, 푓 푢1 = 2푛 + 1 2푖, 푓표푟 2 ≤ 푖 ≤ 푛, 푖 푖푠 푒푣푒푛 푓 푢 = 푖 ≢ 2 푚표푑 3 푖 2푖 − 1, 푓표푟 3 ≤ 푖 ≤ 푛 − 1, 푖 푖푠 표푑푑, 푓 푢푖 = 2푖 + 1 , 푓표푟 1 ≤ 푖 ≤ 푛 − 1, 푖 푖푠 표푑푑 , 푖 ≡ 2 푚표푑 3 ′ 푓 푢푖 = 2푖, 푓표푟 1 ≤ 푖 ≤ 푛 − 1 , 푖 푖푠 표푑푑 " 푓 푢푖 = 2푖 + 1, 푓표푟 1 ≤ 푖 ≤ 푛 − 1, 푖 푖푠 표푑푑 , 푖 ≢ 2 푚표푑 3 " 푓 푢푖 = 2푖 − 1 , 푓표푟 1 ≤ 푖 ≤ 푛 − 1, 푖 푖푠 표푑푑 , 푖 ≡ 2 푚표푑 3 Since푓 푐 =1

gcd 푓 푐 , 푓 푢푖 = 1, 푓표푟 1 ≤ 푖 ≤ 푛 ′ gcd 푓(푢1), 푓 푢1 = gcd 2푛 + 1,2 = 1 " gcd 푓 푢1 , 푓 푢푖 = gcd 2푛 + 1,3 = 1, 푓표푟 푛 ≢ 1 푚표푑 3 Since푛 ≢ 1 푚표푑 3 , 2푛 + 1 ≢ 0 푚표푑 3 " Therefore gcd 푓 푢1 , 푓 푢푖 = (2n+1,3)= 1, Similar to previous theorem for all other pair of adjacent vertices gcd =1 Thus f is a prime labeling. Hence G is a prime graph.

Theorem 2.3 ∗ The graph obtained by duplicating all the rim vertices by edges inCrown 퐶푛 is a prime graph. Proof: ∗ Let푉(퐶푛 ) = {푢푖 ,푣푖 / 1 ≤ 푖 ≤ 푛} ∗ 퐸 퐶푛 ={푢푖 푢푖+1 / 1 ≤ 푖 ≤ 푛} ∪ 푢푖 푣푖 /1 ≤ 푖 ≤ 푛 υ {푢푛 푢1 } ∗ Let G be the graph obtained by duplicating all the rim vertices by edges inCrown 퐶푛 and let the new edges be ′ " ′ " ′ " 푢1푢1, 푢2푢2,… , 푢푛 푢푛by duplicating the vertices 푢1, 푢2 , … . 푢푛 respectively, Then, ′ " 푉 퐺 = { 푢푖 , 푣푖 , 푢푖 , 푢푖 / 1 ≤ 푖 ≤ 푛} ′ ′ " " 퐸 퐺 = 푢푖 푢푖+1/1 ≤ 푖 ≤ 푛 − 1 ∪ {푢푖 푣푖 , 푢푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖 /1 ≤ 푖 ≤ 푛} ∪ { 푢푛 푢1} 푉 퐺 = 4푛, 퐸 퐺 = 5푛. Define a labeling 푓 ∶ 푉 퐺 → 1,2,3, … , 4푛 as follows ′ " Let 푓 푢1 = 1, 푓 푢1 = 2, 푓 푢1 = 3 and푓 푣1 = 4. 푓 푣푖 = 4푖, 푓표푟 2 ≤ 푖 ≤ 푛 푓 푢푖 = 4푖 − 1, 푓표푟 2 ≤ 푖 ≤ 푛, ′ 푓 푢푖 = 4푖 − 3, 푓표푟 2 ≤ 푖 ≤ 푛, " 푓 푢푖 = 4푖 − 2, 푓표푟 2 ≤ 푖 ≤ 푛, Since 푓 푢1 = 1 ′ " gcd 푓(푢1), 푓 푢2 = 1, gcd 푓 푢1 , 푓 푢1 = 1,gcd 푓 푢1 , 푓 푢1 = 1,

gcd 푓(푢1), 푓 푢푛 =1, gcd 푓(푢1), 푓 푣1 = 1. Then

gcd 푓(푢푖 ), 푓 푢푖+1 = gcd 4푖 − 1),4 푖 + 1 − 1 = gcd 4푖 − 1,4푖 + 3 = 1for 2≤ 푖 ≤ 푛

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′ gcd 푓(푢푖 ), 푓 푢푖 = gcd 4푖 − 1,4푖 − 3 = 1for 2≤ 푖 ≤ 푛 as these two numbers are odd and their differences are 4,2 respectively. " gcd 푓(푢푖 ), 푓 푢푖 = gcd 4푖 − 1,4푖 − 2 = 1for 2≤ 푖 ≤ 푛

gcd 푓(푢푖 ), 푓 푣푖 = gcd 4푖 − 1, 4푖 = 1for 2≤ 푖 ≤ 푛 as they are consecutive integers Thus푓 is a prime labeling. Hence G is a prime graph.

Theorem 2.4 The graph obtained by duplicating all the rim vertices by edges in Helm 퐻푛is a prime graph..If 푛 ≢ 4 푚표푑 5

Proof. Let 푉(퐻푛 ) = {푐, 푢푖 ,푣푖 / 1 ≤ 푖 ≤ 푛} 퐸(퐻푛 ) = 푐푢푖 , 푢푖 푣푖 /1 ≤ 푖 ≤ 푛 ∪ {푢푖 푢푖+1 / 1 ≤ 푖 ≤ 푛 − 1 υ 푢푛 푢1 } Let G be the graph obtained by duplicating all the rim vertices by edgesin Helm 퐻푛and let the new edges ′ " ′ " ′ " be 푢1푢1, 푢2푢2,… , 푢푛 푢푛by duplicating the rim vertices 푢1, 푢2 , … . 푢푛 respectively, Then, ′ " ′ " ′ " 푉 퐺 = {푐, 푢푖 , 푣푖 ,푢푖 , 푢푖 / 1 ≤ 푖 ≤ 푛}퐸 퐺 = 푐푢푖 , 푢푖 푣푖 ,푢푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖 /1 ≤ 푖 ≤ 푛 ∪ 푢푖 푢푖+1/1 ≤ 푖 ≤ 푛 − 1∪ 푢푛푢1}. 푉퐺=4푛+1, 퐸퐺=6푛, Define a labeling 푓 ∶ 푉 퐺 → 1,2,3,… , 4푛 + 1 as follows ′ " Let푓 푐 = 1 , 푓 푢1 = 5, 푓 푢1 = 3, 푓 푢1 = 4, and푓 푣1 = 2. 푓 푢푖 = 4푖 − 1, 푓표푟 2 ≤ 푖 ≤ 푛, ′ 푓 푢푖 = 4푖, 푓표푟 2 ≤ 푖 ≤ 푛, " 푓 푢푖 = 4푖 + 1, 푓표푟 2 ≤ 푖 ≤ 푛, 푓 푣푖 = 4푖 − 2, 푓표푟 2 ≤ 푖 ≤ 푛 Since 푓 푐 = 1

gcd 푓(푐), 푓 푢푖 = 1 ′ gcd 푓 푢1 , 푓 푢1 =gcd(5,3) = 1 " gcd 푓 푢1 , 푓 푢1 = gcd 5,4 = 1

gcd 푓(푢1), 푓 푣1 = gcd(5,2)= 1 gcd 푓(푢1), 푓 푢2 = gcd 5,7 =1 gcd 푓(푢푛 ), 푓 푢1 = gcd 4n − 1,5 =1 Since 푛 ≢ 4 푚표푑 5 and 4n-1 is not a multiple of 5 gcd 푓(푢푖 ), 푓 푢푖+1 = gcd 4푖 − 1),4 푖 + 1 − 1 = gcd 4푖 − 1,4푖 + 3 = 1for 2≤ 푖 ≤ 푛 " gcd 푓(푢푖 ), 푓 푢푖 = gcd 4푖 − 1,4푖 + 1 = 1for 2≤ 푖 ≤ 푛 as these two numbers are odd and also their differences are 4,2 respectively. ′ gcd 푓(푢푖 ), 푓 푢푖 = gcd 4푖 − 1,4푖 = 1for 2≤ 푖 ≤ 푛 gcd 푓(푢푖 ), 푓 푣푖 = gcd 4푖 − 1,4푖 − 2 = 1for 2≤ 푖 ≤ 푛 as they are consecutive integers Thus푓 is a prime labeling. Hence G is a prime graph.

Theorem 2.5 The graph obtained by duplicating all the rim vertices by edges in Gear퐺푛 is a prime graph..If 푛 ≢ 4 푚표푑 5 Proof. Let 푉(퐺푛 )= {푐, 푢푖 ,푣푖 / 1 ≤ 푖 ≤ 푛} 퐸(퐺푛 ) = 푐푣푖 , 푣푖 푢푖 /1 ≤ 푖 ≤ 푛 ∪ {푢푖 푢푖+1 / 1 ≤ 푖 ≤ 푛 − 1 υ 푢푛 푢1 } Let G be the graph obtained by duplicating all the rim vertices by edgesin Gear 퐺푛 and let the new edges be ′ " ′ " ′ " 푢1푢1, 푢2푢2,… , 푢푛 푢푛by duplicating the rim vertices 푢1, 푢2 , … . 푢푛 respectively, Then,

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′ " ′ " ′ " 푉 퐺 = {푐, 푢푖 , 푢푖 , 푢푖 , 푣푖 / 1 ≤ 푖 ≤ 푛}퐸 퐺 = 푐푣푖 , 푣푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖 /1 ≤ 푖 ≤ 푛 ∪ 푢푖 푢푖+1/1 ≤ 푖 ≤ 푛 − 1∪ 푢푛푢1}.푉퐺=4푛+1, 퐸퐺=6푛. Define a labeling 푓 ∶ 푉 퐺 → 1,2,3, … , 4푛 + 1 as follows ′ " Let푓 푐 = 1, 푓 푢1 = 5, 푓 푢1 = 3, 푓 푢1 = 4, and 푓 푣1 = 2. 푓 푢푖 = 4푖 − 1, 푓표푟 2 ≤ 푖 ≤ 푛, ′ 푓 푢푖 = 4푖, 푓표푟 2 ≤ 푖 ≤ 푛, " 푓 푢푖 = 4푖 + 1, 푓표푟 2 ≤ 푖 ≤ 푛, 푓 푣푖 = 4푖 − 2, 푓표푟 2 ≤ 푖 ≤ 푛 Since 푓 푐 = 1

gcd 푓(푐), 푓 푣푖 = 1, 푓표푟 1 ≤ 푖 ≤ 푛 Similar to the previous theorem we can show that for all other pair of adjacent vertices g.c.d is 1 Thus푓 is a prime labeling. Hence G is a prime graph.

Theorem 2.6 The graph obtained by duplicating the centre vertex andall the alternate vertices by edges in Star 푆푛 = 퐾1,푛 is a prime graph.. If n is even

Proof. Let푉(푆푛 ) = {푐, 푢푖 / 1 ≤ 푖 ≤ 푛} 퐸(푆푛 )= 푐푢푖 /1 ≤ 푖 ≤ 푛 Let G be the graph obtained by duplicating the centre vertex and all the alternate vertices by edgesin Star ′ " ′ " ′ " ′ " 푆푛 = 퐾1,푛 and let the new edges be 푐 푐 ,푢1푢1, 푢3푢3, … , 푢푛−1푢푛−1by duplicating the vertices 푐, 푢1, 푢3 , … . 푢푛−1respectively, Then, ′ " ′ " ′ " ′ " 푉 퐺 = {푐, 푐 , 푐 , 푢푖 / 1 ≤ 푖 ≤ 푛} ∪ 푢푖 , 푢푖 /1 ≤ 푖 ≤ 푛, 푖 푖푠 표푑푑 퐸 퐺 = 푐푢푖 , 푐푐 , 푐푐 , 푐 푐 /1 ≤ 푖 ≤ 푛 ∪ ′ " ′ " 푢푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖 /1 ≤ 푖 ≤ 푛, 푖 푖푠 표푑푑 . 5푛 푉 퐺 = 2푛 + 3, 퐸 퐺 = +3 2 Define a labeling 푓 ∶ 푉 퐺 → 1,2,3,… , 2푛 + 3 as follows 2푖 + 1, 푓표푟 1 ≤ 푖 ≤ 푛, 푖 푖푠 표푑푑 Let푓 푐 = 1, 푓 푐′ = 2푛 + 2, 푓 푐" = 2푛 + 3. 푓(푢 ) = 푖 2푖 − 2, 푓표푟 2 ≤ 푖 ≤ 푛, 푖 푖푠 푒푣푒푛, ′ 푓 푢푖 = 2푖 + 2, 푓표푟 1 ≤ 푖 ≤ 푛, 푖 푖푠 표푑푑 " 푓 푢푖 = 2푖 + 3, 푓표푟 1 ≤ 푖 ≤ 푛, 푖 푖푠 표푑푑 Since 푓 푐 = 1

gcd 푓 푐 , 푓 푢푖 = 1, 푓표푟 1 ≤ 푖 ≤ 푛 gcd 푓 푐 , 푓 푐′ = 1, gcd 푓 푐 , 푓 푐" = 1 gcd 푓 푐′ , 푓 푐" = gcd 2푛 + 2,2푛 + 3 = 1 ′ gcd 푓(푢푖 ), 푓 푢푖 = gcd 2푖 + 1,2푖 + 2 = 1for 1≤ 푖 ≤ 푛 ′ " gcd 푓(푢푖 ), 푓 푢푖 = gcd 2푖 + 2,2푖 + 3 = 1for 1≤ 푖 ≤ 푛 " gcd 푓(푢푖 ), 푓 푢푖 = gcd 2푖 + 1,2푖 + 3 = 1for 1≤ 푖 ≤ 푛 as they are consecutive odd integers Thus푓 is a prime labeling. Hence G is a prime graph.

Theorem 2.7 The graph obtained by duplicating all the alternate vertices by edges in Friendship graph 푇 푛 , except centre,is a prime graph.

Proof. Let 푉(푇푛 ) = {푐, 푢푖 / 1 ≤ 푖 ≤ 2푛} 퐸(푇푛 ) = 푐푢푖 /1 ≤ 푖 ≤ 2푛 ∪ {푢푖 푢푖+1 / 1 ≤ 푖 ≤ 2푛 − 1, 푖 is odd} . Let G be the graph obtained by duplicating all the alternate vertices by edgesin 푇푛 and let the new edges be ′ " ′ " ′ " 푢1푢1, 푢3푢3,… , 푢2푛−1푢2푛−1by duplicating the vertices푢1, 푢3 , … . 푢2푛−1respectively, Then Manuscript id. 249778 www.ijstre.com Page 13 Some New Prime Graphs

′ " 푉(퐺) = {푐, 푢푖 ,푢푖 , 푢푖 / 1 ≤ 푖 ≤ 2푛} ′ " ′ " 퐸 퐺 = 푐푢푖 /1 ≤ 푖 ≤ 2푛 ∪ {푢푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖+1 / 1≤i≤2n-1,iis odd} 푉 퐺 = 4푛 + 1, 퐸 퐺 = 6푛, Define a labeling 푓 ∶ 푉 퐺 → 1,2,3,… , 4푛 + 1 as follows Let 푓 푐 = 1. 2푖 + 1, 푓표푟 1 ≤ 푖 ≤ 2푛 − 1, 푖 푖푠 표푑푑 푓 푢 = 푖 2푖 − 2, 푓표푟 2 ≤ 푖 ≤ 2푛, 푖 푖푠 푒푣푒푛 ′ 푓 푢푖 = 2푖 + 2, 푓표푟 1 ≤ 푖 ≤ 2푛 − 1, 푖 푖푠 표푑푑 " 푓 푢푖 = 2푖 + 3, 푓표푟 1 ≤ 푖 ≤ 2푛 − 1, 푖 푖푠 표푑푑 Since 푓 푐 = 1

gcd 푓 푐 , 푓 푢푖 = 1, 푓표푟 1 ≤ 푖 ≤ 2푛 − 1, 푖 푖푠 표푑푑 gcd 푓 푐 , 푓 푢푖 = 1, 푓표푟 2 ≤ 푖 ≤ 2푛, 푖 푖푠 푒푣푒푛 If i is odd

gcd 푓 푢푖 , 푓 푢푖+1 = gcd 2푖 + 1,2(푖 + 1 − 2) = gcd 2푖 + 1,2푖 = 1 for 1≤ 푖 ≤ 2푛 − 1 ′ gcd 푓 푢푖 , 푓 푢푖 = gcd 2푖 + 1,2푖 + 2 = 1 for 1≤ 푖 ≤ 2푛 − 1 " gcd 푓(푢푖 ), 푓 푢푖 = gcd 2푖 + 1,2푖 + 3 = 1for 1≤ 푖 ≤ 2푛 − 1 ′ " .gcd 푓(푢푖 ), 푓 푢푖 = gcd 2푖 + 2,2푖 + 3 = 1for 1≤ 푖 ≤ 2푛 − 1 Thus fis a prime labeling Hence G is a prime graph.

Theorem 2.8 The graph obtained by duplicating all the alternate vertices by edges in Prism 퐷푛 is a prime graph.Where n is even.

Proof. Let푉(퐷푛 ) = {푢푖 / 1 ≤ 푖 ≤ 푛} 퐸(퐷푛 ) = {푢푖 푢푖+1 / 1 ≤ 푖 ≤ 푛 − 1 υ 푢푛 푢1 }υ 푢1푢푖 / 3 ≤ 푖 ≤ 푛 − 1} Let G be the graph obtained by duplicating all the alternate vertices by edges in Prime 퐷푛 and let the new ′ " ′ " ′ " edges be 푢1푢1,푢3푢3, … , 푢푛−1푢푛−1by duplicating the vertices 푢1, 푢3 , … . 푢푛−1respectively, ′ " 푉 퐺 = {푢푖 / 1 ≤ 푖 ≤ 푛} ∪ {푢푖 , 푢푖 / 1 ≤ 푖 ≤ 푛 − 1, 푖 푖푠 표푑푑}퐸 퐺 = ′ " ′ " 푢푖 푢푖+1/1 ≤ 푖 ≤ 푛 − 1 ∪ {푢푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖 /1 ≤ 푖 ≤ 푛 − 1, 푖푖푠표푑푑} ∪ 푢푛 푢1 ∪ {푢1푢푖 /3 ≤ 푖 ≤ 푛 − 1 } . 푉 퐺 = 3푛, 퐸 퐺 = (7푛 − 6)/2 . Let 푓 푢1 = 1. 2푖, 푓표푟 2 ≤ 푖 ≤ 푛, 푖 푖푠 푒푣푒푛 푓(푢 )= 푖 2푖 − 1, 푓표푟 3 ≤ 푖 ≤ 푛 − 1, 푖 푖푠 표푑푑, 푖 ≢ 2 푚표푑 3 , 푓 푢푖 = 2푖 + 1 , 푓표푟 3 ≤ 푖 ≤ 푛 − 1, 푖 푖푠 표푑푑 , 푖 ≡ 2 푚표푑 3 ′ 푓 푢푖 = 2푖, 푓표푟 1 ≤ 푖 ≤ 푛 , 푖 푖푠 표푑푑 " 푓 푢푖 = 2푖 + 1, 푓표푟 1 ≤ 푖 ≤ 푛, 푖 푖푠 표푑푑 , 푖 ≢ 2 푚표푑 3 " 푓 푢푖 = 2푖 − 1 , 푓표푟 1 ≤ 푖 ≤ 푛, 푖 푖푠 표푑푑 , 푖 ≡ 2 푚표푑 3 Since 푓 푢1 = 1 gcd 푓(푢1), 푓 푢푖 = 1for 3 ≤i≤ n-1 Similar to the theorem 2.1 we canshow that for all other pair of adjacent vertices g.c.dis 1 Thus f isa prime labeling Hence G is a prime graph.

Theorem 2.9 The graph obtained by duplicating all the vertices of the cycles by edgesin Butterfly graph퐵푛,푚 is a prime graph. If m ≥ 2n-2 Proof.

Let푉(퐵푛,푚 ) = {푢푖 / 1 ≤ 푖 ≤ 2푛 − 1} ∪ {푣푖 / 1 ≤ 푖 ≤ 푚}퐸 퐵푛,푚 ={푢푖 푢푖+1 /1 ≤ 푖 ≤ 푛 − 1 υ{푢푖 푢푖+1 / (푛 + 1) ≤ 푖 ≤ 2푛 − 2}υ {푢푛 푢1 , 푢1푢푛+1, 푢2푛−1푢1} ∪ {푢1푣푖 /1 ≤ 푖 ≤ 푚 }

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Case (i). If m = 2n -2 Let G be the graph obtained by duplicating all the vertices of two copies of 퐶푛 by edgesin 퐵푛,푚 and let the ′ " ′ " ′ " new edges be 푢1푢1,푢2푢2, … , 푢2푛−1푢2푛−1by duplicating the vertices 푢1, 푢2 , … . 푢2푛−1respectively, Then, ′ " 푉 퐺 = {푢푖 , 푢푖 , 푢푖 / 1 ≤ 푖 ≤ 2푛 − 1} ∪ 푣푖 / 1 ≤ 푖 ≤ 푚 퐸 퐺 = 푢푖 푢푖+1/1 ≤ 푖 ≤ 푛 − 1 ∪ ′ " ′ " 푢푖 푢푖+1/ 푛 + 1 ≤ 푖 ≤ 2푛 − 2 ∪ 푢푖 푢푖 , 푢푖 푢푖 , 푢푖 푢푖 /1 ≤ 푖 ≤ 2푛 − 1 ∪ 푢푛 푢1,푢1푢푛+1, 푢2푛−1푢1 ∪ { 푢1푣푖 /1 ≤ 푖 ≤ 푚}. 푉 퐺 = 6푛 − 3 + 푚, 퐸 퐺 = 8푛 − 3 + 푚. Define a labeling 푓 ∶ 푉 퐺 → 1,2,3,… ,6푛 − 3 + 푚 as follows Let푓 푢1 = 1 푓 푢푖 = 4푖 − 3, 푓표푟 1 ≤ 푖 ≤ 푛 푎푛푑 푛 + 1 ≤ 푖 ≤ 2푛 − 1 ′ 푓 푢푖 = 4푖 − 2, 푓표푟 1 ≤ 푖 ≤ 2푛 − 1, " 푓 푢푖 = 4푖 − 1, 푓표푟 1 ≤ 푖 ≤ 2푛 − 1, 푓 푣푖 = 4푖, 푓표푟 1 ≤ 푖 ≤ 푚 (= 2푛 − 2) Since 푓 푢1 = 1 gcd 푓(푢1), 푓 푣푖 = 1 푓표푟 1 ≤ 푖 ≤ 푚. gcd 푓 푢1 , 푓 푢푛 = 1 gcd 푓 푢1 , 푓 푢푛+1 = 1 gcd 푓(푢1), 푓 푢2푛−1 = 1 gcd 푓(푢푖 ), 푓 푢푖+1 = gcd 4푖 − 3),4 푖 + 1 − 3 = gcd 4푖 − 3,4푖 + 1 = 1for 1≤ 푖 ≤ 푛-1 and n+1≤ 푖 ≤ 2푛-2 " gcd 푓(푢푖 ), 푓 푢푖 = gcd 4푖 − 3,4푖 − 1 = 1for 1≤ 푖 ≤ n-1 and n+1≤ 푖 ≤ 2푛-2 as these two numbers are odd and their differences are 4,2 respectively. ′ gcd 푓(푢푖 ), 푓 푢푖 = gcd 4푖 − 3,4푖 − 2 = 1for 1≤ 푖 ≤ n-1 and n+1≤ 푖 ≤ 2푛-2 Then f is a prime labeling.

Case (ii).If m > 2n -2 Then the same labeling given in case (i) can be given for all the vertices up to m=2n-2. For all other vertices give consecutive labelsso that the resulting labeling is prime. Thus f is a prime labeling Hence G is a prime graph.

Examples

Illustration 2.1

u1' 3 u1ʺ 2 1

u1

12 4

9 11 5 6

8 10 7

Figure1. Prime labeling of duplication of all the alternate vertices by edges in 퐶6

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Illustration 2.2

u1' u1ʺ 2 3 17

u1 16 4 15 6 13 c 5 1 14 7 11 12 8

9 10

Figure1. Prime labeling of duplication of all the alternate rim vertices by edges in 퐶8 .

Illustration 2.3

u1' u1ʺ 3 2 1

u1 4 5 18 v 1 7

8 6 17 19 20

16 12

11 9 14 15

10 13

∗ Figure1. Prime labeling of duplication of all the rim vertices by edges in Crown 퐶5 Illustration 2.4

u1' u1ʺ 3 4 5

u1

21 8 18 v1 2 7 c 20 19 9 14 1 6

10 11 12 17 15

16 13

Figure1. Prime labeling of duplication of all the rim vertices by edges in Helm퐻5 Manuscript id. 249778 www.ijstre.com Page 16 Some New Prime Graphs

Illustration 2.5

u1' u1ʺ 3 4 5

u1

21 8 v1 2 7 18 19 c 20 6 9 1 14 10 11 12 17 15

16 13

Figure1. Prime labeling of duplication of all the rim vertices by edges in Gear graph퐺 5.

Illustration 2.7

u ' 1 u1ʺ 4 5

3 u1 u2 10 2 c 13 1 7 11 8 12 6

9

Figure1. Prime labeling of duplication of all the alternate vertices by edges in

Friendship graph 푇3 , except centre .

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Illustration 2.8

u1' u1ʺ 2 3 1

u1 16 4 15 6 5 13

14 7 11 12 8

9 10

Figure1. Prime labeling of duplication of all the alternate vertices by edges inPrism 퐷8

Illustration 2.9 8 12 4 16 18 23 19 22 26 15 17 21 25 27 14 13 u1ʺ 3 u1 1 u1' 2 11 9 29 30 5 33 35 31 10 6 7 34 20 32 24 28

Figure1. Prime labeling of duplication of all the vertices of two cycles 퐶5 by edges in Butterfly graph 퐵5,8 .

III. CONCLUSION Here we investigate Nine corresponding results on prime labeling analogues work can be carried out for other families also.

REFERENCES [1.] Bondy.J.A and Murthy. U.S.R, “ and Application”. (North - Holland). Newyork (1976). [2.] Deretsky.T, Lee.S.M and Mitchem.J, “On Vertex Prime Labeling of Graphs inGraph Theory, Combinatorics and Applications”, Vol. 1 Alavi.J, Chartrand. G, Oellerman.O and Schwenk. A, eds. Proceedings 6th International Conference Theory and Application of Graphs (Wiley, New York 1991) 359-369. [3.] Fu. H.C and Huany. K.C. “On Prime Labeling Discrete Math”, 127 (1994) 181-186. [4.] Lee. S. M, Wui.L and Yen.J“On the Amalgamation of Prime Graphs”, Bull. Malaysian Math. Soc. (Second Series) 11, (1988) 59-67. [5.] SundaramM.Ponraj&Somasundaram. S. (2006) “On Prime Labeling Conjecture”, ArsCombinatoria, 79, 205-209. [6.] Tout. A, Dabboucy.A.N and Howalla. K, “Prime Labeling of Graphs”, Nat.Acad.Sci letters11 (1982)365-368 Combinatories and Application Vol.1Alari.J(Wiley.N.Y 1991) 299-359. [7.] S.K Vaidya and LekhaBijukumar, “Some New Families of Mean Graphs”, Journal of Mathematics Research, vol.2. No.3; August 2010.

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