Signed Product Cordial labeling in duplicate graphs of Bistar, Double Star and Triangular Ladder Graph

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1 Signed Product Cordial labeling in duplicate graphs of Bistar Double Star Triangular Ladder Graph P.P Ulaganathan #1 B. Selvam #2 P. Vijaya kumar #3 12 Department of Mathematics S.I.V.E.T College Gowrivakkam Chennai India. 3 Department of Mathematics Sriram Engineering College Perumalpattu Tiruvallur India. Abstract -In this paper we prove that the duplicate graph of triangular ladder extended duplicate graph of bistar extended duplicate graph of double star admitsigned product cordial Keywords -Graph labeling duplicate graph triangular ladder bistar double star signed product cordial AMS Subject Classification 05C78 I. INTRODUCTION The concept of graph labeling was introduced by Rosa in 1967 [6]. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition(s). If the domain of the mapping is the set of vertices (or edges) then the labeling is called a vertex labeling (or an edge labeling). In the intervening years various labeling of graphs have been investigated in over 2000 papers [4]. The concept of cordial labeling was introduced by I. Cahit[2]. The concept of duplicate graph was introduced by E. Sampathkumar he proved many results on it [7]. The concept of signed product cordial labeling was introduced by J. BaskarBabujee he proved that many graphs admit signed product cordial K. Thirusangu P.P. Ulaganathan B. Selvam have proved that the duplicate graph of a path graph is Cordial [8].K. Thirusangu P.P. Ulaganathan P. Vijayakumar have proved that the duplicate graph of Ladder graph is cordial total cordial prime cordial[9]. II. PRELIMINARIES In this section we give the basic notions relevant to this paper. Definition 2.1 A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition(s). If the domain of the mapping is the set of vertices (or edges) then the labeling is called a vertex labeling (or an edge labeling). Definition2.3 Let be a simple graph. A duplicate graph of G is where the vertex set is bijective( we write ) the edge set of DG is defined as The edge uv is in E if only if both are edges in. Definition2.4A vertex labeling of graph G with induced edge labeling defined by is called a signed product cordial labeling if where is the number of vertices labeled with -1 is the number of vertices labeled with 1 is the number of edges labeled with is the number of edges labeled with 1. [1]. Definition 2.5 The ladder graph is a planar undirected graph with 2m vertices 3m 2 edges. It is obtained as the cartesian product of two path graphs one of which has only one edge where m is the number of rungs in the ladder. Definition 2.6 The triangular ladder graph T is obtained from the ladder graph by joining Definition 2.7 The Star graph is a complete bipartite graph K 1 m where m represents the number of vertices S m has (m 1) edges. Definition 2.8 Double star is a tree obtained from the star by adding a new pendent ISSN http// Page 19

2 edge of the existing m pendant vertices. It has 2m + 1 vertices 2m edges.[11]. Definition 2.9 Bistar is a graph obtained from by joining m pendant edges to each end of. The edge is called the central edge of the vertices of are called the central vertices of [5]. Definition2.10 The extended duplicate graph of bistar denoted by is obtained from the duplicate graph of bistar by joining. Definition2.11 The extended duplicate graph of double star denoted by is obtained from the duplicate graph of double star by joining. III. MAIN RESULTS Algorithm 3.1 Construction of duplicate graph of triangular ladder Illustration Remark The Duplicate graph of Triangular ladder has 2m vertices 8m 6 edges. Algorithm3.2 Assignment of labels to vertices Theorem 3.1 The duplicate graph of triangular ladder admits signed product cordial ProofLet be the set of vertices edges of the duplicate graph of triangular ladder. Using the algorithm 3.2 each of the 2m vertices receive label 1 1. Using the induced function defined by The 4m 3 edges namely receive label 1 the 4m 3 edges namely receive label 1 respectively. Thus the number of edges labeled with 1 1 differ at most by one. Hence the duplicate graph of triangular ladder admits signed product cordial Fig. 1Example of Triangular ladder duplicate graph its Fig 2. Example of signed product cordial labeling in ISSN http// Page 20

3 Algorithm 3.3 (Construction of extended duplicate graph of Bistar ). Let be the set of vertices edges of the. Case (i) When m is odd Using the algorithm 3.4 each of the 2m + 2 vertices receive label 1 1. Using the induced function defined in theorem3.1 the 2m + 2 edges namely receive label 1 the 2m + 1 edges namely receive label 1. Thus the number of edges labeled with 1 1 differ at most by one. Case (ii) When m is even Using the algorithm 3.4 each of the 2m + 2 vertices receive label 1 1.Using the induced function defined in theorem3.1 the 2m + 2 edges namely receive label 1 the 2m + 1 edges namely receive label 1.Thus the number of edges labeled with 1 1 differ at most by one. Hence the extended duplicate graph of the bistar admits signed product cordial Fig.3 Example of bistar Remark TheExtended duplicate graph of Bistar has 4m + 4 vertices 4m+3 edges. Algorithm 3.4 Assignment of labels to vertices Theorem 3.2 The extended duplicate graph of the bistar admits signed product cordial Proof Fig 4 Example of signed product cordial labeling in Algorithm 3.5 (Construction of extended duplicate graph of double star ) ISSN http// Page 21

4 Fig 5 Example of double star Algorithm 3.6 (Assignment of labels to vertices) ISSN http// Page 22

5 Theorem 3.3The extended duplicate graph of double star admits signed product cordial Proof Let vertices the set of edges of. be the set of Using the algorithm 3.6 each of the 2m + 1 vertices receive label 1 1. Using the induced function as in theorem 3.1 the 2m edges namely receive label 1 the 2m + 1 edges namely receive label 1 respectively. Thus the number of edges labeled with 1 1 differ by one. receive label 1 the 2m + 1 edges namely receive label 1. Thus the number of edges labeled with 1 1 differ by one. Hence the extended duplicate graph of double star admits signed product cordial Using the algorithm 3.6 each of the 2m + 1 vertices receive label 1 1. Using the induced function as in theorem 3.1 the 2m edges namely receive label 1 the 2m + 1 edges namely receive label 1.Thus the number of edges labeled with 1 1 differ by one. Using the algorithm 3.6 each of the 2m + 1 vertices receive label 1 1. Using the induced function defined as in theorem 3.1 the 2m edges namely receive label 1 the 2m + 1 edges namely. receive label 1. Thus the number of edges labeled with 1 1 differ by one. Using the algorithm 3.6 each of the 2m + 1 vertices receive label 1 1. Using the induced function as in theorem 3.1 the 2m edges namely Fig 6 Example of signed product cordial labeling in ISSN http// Page 23

6 IV. Conclusion We proved that the duplicate graph of triangular ladder the extended duplicate graph of bistar the extended duplicate gtraph of the double star admit signed product cordial V. References [1] J.Baskar Babujee L. Shobana On Signed Product Cordial Labeling Applied Mathematics [2] I. Cahit On cordial 3-equitable labeling of graphs Utilitas. Math. 37(1990) [3] I. Cahit Cordial graphs a weaker version of graceful harmonious graphs ArsCombinatorica 23 (1987) [4] J. A. Gallian A Dynamic Survey of Graph Labeling The Electronic Journal of Combinatorics #DS6 (2014). [5] P. Jeyanthi A. Maheswari One Modulo Three Mean Labeling of Graphs American Journal of Applied Mathematics Statistics 2014 Vol.2 N [6] A. Rosa On certain valuations of the vertices of a graph Theory of graphs (Internat. Symposium Rome July 1996) Gordon Breach N. Y. Dunod Paris (1967) [7] E. Sampathkumar On duplicate graphs Journal of the Indian Math. Soc. 37 (1973) [8] K. Thirusangu P.P. Ulaganathan B. Selvam Cordial labeling in duplicate graphs Int. J. Computer Math. Sci. Appl. 4(1-2) (2010) [9] K. Thirusangu P.P. Ulaganathan P. Vijaya kumar Some cordial labeling of duplicate graph of ladder graph Annals of Pure Applied Mathematics 8(2) (2014) [10] S.K. Vaidya N.H. Shah 3-Equitable Labeling for Some Star Bistar Related Graphs International Journal of Mathematics Scintific Computing (ISSN ) Vol. 2 No [11] M. Venkatachalam J. Vernold Vivin K. Kaliraj armonious Colouring on double star graph families Tamkang Journal of Mathematics Vol. 43 No [12] P. Vijayakumar P.P. Ulaganathan K. Thirusangu Some Cordial Labeling in Extended Duplicate Graph of Star Graph International Journal of Applied Engineering Research ISSN Vol. 10 No.80 (2015) [13] S. Meena P. Kavitha Prime Labeling of Duplication of Some Star related Graphs International Journal of Mathematics Trends Technology- Volume23 Number1 July [14] A. Nellai Murugan Near Meanness on Special Types of Graphs International Journal of Mathematics Trends Technology Volume 10 Number 1 Jun [15] S.S.Shya S. Somasundaram New Classes of Harmonic Mean Graphs International Journal of Mathematics Trends Technology- Volume22 Number1 June ISSN http// Page 24