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Some New Prime Graphs 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 , Friendship graph 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 vertex 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. Manuscript id. 249778 www.ijstre.com Page 10 Some New Prime Graphs 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.
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  • Some New Families of 4-Prime Cordial Graphs
    International J.Math. Combin. Vol.3(2017), 125-135 Some New Families of 4-Prime Cordial Graphs R.Ponraj (Department of Mathematics, Sri Paramakalyani College, Alwarkurichi-627412, India) Rajpal Singh and R.Kala (Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627012, India) E-mail: [email protected], [email protected], [email protected] Abstract: Let G be a (p, q) graph. Let f : V (G) 1, 2,...,k be a function. For → { } each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if vf (i) vf (j) 1, i, j 1, 2,...,k and ef (0) ef (1) 1 where vf (x) denotes the | − | ≤ ∈ { } | − | ≤ number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with admits a k-prime cordial labeling is called a k-prime cordial graph. In this paper we investigate 4-prime cordial labeling behavior of shadow graph of a path, cycle, star, degree splitting graph of a bistar, jelly fish, splitting graph of a path and star. Key Words: Cordial labeling, Smarandachely cordial labeling, cycle, star, bistar, splitting graph. AMS(2010): 05C78. §1. Introduction In this paper graphs are finite, simple and undirected. Let G be a (p, q) graph where p is the number of vertices of G and q is the number of edge of G. In 1987, Cahit introduced the concept of cordial labeling of graphs [1]. Sundaram, Ponraj, Somasundaram [5] have been introduced the notion of prime cordial labeling and discussed the prime cordial labeling behavior of various graphs.
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  • The Structure of Graphs with Circular Flow Number 5 Or More, and The
    The structure of graphs with Circular flow number 5 or more, and the complexity of their recognition problem L. Esperet ∗ G. Mazzuoccolo y M. Tarsi z September 10, 2015 Abstract For some time the Petersen graph has been the only known Snark with circular flow number 5 (or more, as long as the assertion of Tutte's 5-flow Conjecture is in doubt). Although infinitely many such snarks were presented eight years ago in [9], the variety of known methods to construct them and the structure of the obtained graphs were still rather limited. We start this article with an analysis of sets of flow values, which can be transferred through flow networks with the flow on each edge restricted to the open interval (1; 4) modulo 5. All these sets are symmetric unions of open integer intervals in the ring R=5Z. We use the results to design an arsenal of methods for constructing snarks S with circular flow number φc(S) 5. As one indication to the ≥ diversity and density of the obtained family of graphs, we show that it is sufficiently rich so that the corresponding recognition problem is NP-complete. arXiv:1501.03774v2 [math.CO] 9 Sep 2015 Keywords: Snarks, Circular flows, Nowhere-zero flows, NP-Completeness. ∗Laboratoire G-SCOP (Grenoble-INP, CNRS), Grenoble, France. Partially supported by ANR Project Heredia (anr-10-jcjc-0204-01), ANR Project Stint (anr-13-bs02- 0007), and LabEx PERSYVAL-Lab (anr-11-labx-0025). yUniversit`adi Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Italy e-mail: [email protected] zThe Blavatnik School of Computer Science, Tel Aviv University, Israel e-mail: [email protected] 1 1 Introduction For an integer k, a nowhere-zero k-flow in a graph G is a flow in some orientation of G, such that the flow value on each arc is in 1; 2; : : : ; k 1 .
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  • Zero Forcing and Power Domination in Graphs
    ZERO FORCING AND POWER DOMINATION IN GRAPHS SUDEEP STEPHEN Master of Science (Mathematics)(University of Madras) School of Mathematical and Physical Sciences The University of Newcastle, Australia A thesis submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy October 2017 Statements Statement of Originality The thesis contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to the final version of my thesis being made available worldwide when deposited in the University's Digital Repository, subject to the provisions of the Copyright Act 1968. Statement of Authorship I hereby certify that the work embodied in this thesis contains published papers of which I am a joint author. I have included as part of the thesis a written statement, endorsed by my supervisor, attesting to my contribution to the joint publication. Signed: SUDEEP STEPHEN The candidate has six papers, of which he is a joint author, embodied in the thesis. As principal supervisor I can attest that the candidate's contribution, in all cases were at least 80% of the final article. Signed: Dr JOE RYAN iv List of Publications Publications Arising from this Thesis (1) D. Ferrero, T. Kalinowski and S. Stephen. Zero forcing in iterated line digraphs. Submitted to Discrete Applied Mathematics, 2017 [37]. (2) R. Davila, T. Kalinowski and S.
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  • Fuzzy Graceful Labeling and Fuzzy Antimagic Labeling of Path and Butterfly Graph
    IAETSD JOURNAL FOR ADVANCED RESEARCH IN APPLIED SCIENCES ISSN NO: 2394-8442 FUZZY GRACEFUL LABELING AND FUZZY ANTIMAGIC LABELING OF PATH AND BUTTERFLY GRAPH R.Jebesty Shajila#1, S.Vimala*2 Research Scholar, Department of Mathematics, Mother Teresa Women’s University, Kodaikanal11, Assistant professor, Department of Mathematics, Mother Teresa Women’s University, Kodaikanal2, E-mail:[email protected], E-mail:[email protected] Abstract - Fuzzy antimagic labeling introduced in this paper. Examine results in vertex antimagic labeling and edge antimagic labeling for some graphs like path graph and butterfly graph. Fuzzy graceful labelling introduced [9].Properties are defined for path graph and butterfly graph through Fuzzy graceful labelling. Key words - Fuzzy labelling, Graceful labelling, fuzzy antimagic path graph, fuzzy antimagic butterfly graph. AMS Mathematics subject classification: 05C,94D I. Introduction A magic square is an arrangement of numbers into a square such that the sum of each row , column and diagonal are equal. The term antimagic then comes from being the opposite of magic or arranging numbers in a way such that no two sums are equal[3]. The interest in graph labelings can trace its roots back to a paper [4] by A.Rosa in 1966. Nora Hartsfield and Gerhard Ringel [5] introduced the concept of antimagic labelling which is an assignment of distinct values to different objects in a graph such a way that when taking certain sums of the labels the sums will all be different. Bodendiek and Walter [6] defined the concept of an antimagic labeling as an edge labelling in which the vertex values form an arithmetic progression starting from a and have common difference d.
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