International Journal of Fuzzy Mathematics and Systems. ISSN 2248-9940 - PDF document

International Journal of Fuzzy Mathematics and Systems. ISSN 2248-9940 Volume 2, Number 1 (2012), pp. 83-94 © Research India Publications Gracefulness of a New Class of Stars Merged with Trees A. Solairaju and S. Ambika Associate Professor of Mathematics, , Trichy-20, India E-mail: solairama@yahoo.co.in Assistant Professor of Mathematics, Government Arts College, Coimbatore-18, India E-mail: ambisadha@yahoo.com Abstract Given a graph G with q edges, a labeling of the nodes with distinct integers from the set {0, 1, 2,…,q} induces an edge labeling where the label of an edge is the absolute difference of the labels of the two nodes incident to that edge. Such a labeling is graceful if the edge labels are distinct. A graph G is called graceful if there exist a graceful labeling of G. In this paper, the gracefulness of a new class of stars merged with trees is obtained. Keywords: graceful graph, Path, Cycle, Trees. AMS classification number: 2000 MSC: 05C78 Introduction Graphs labeling, where the vertices are assigned values subject to certain conditions, have often been motivated by practical problems. Labeled graphs serves as useful mathematical models for a broad range of applications such as Coding theory, including the design of good radar type codes, synch-set codes, missile guidance codes and convolution codes with optimal autocorrelation properties. They facilitate the optimal nonstandard encoding of integers. When studying graceful labeling, only simple graph is considered. A graceful labeling of an undirected graph edges is a one-to-one function from the set of vertices to the set {0, 1, 2,…,q}such that the induced edge labels are all distinct. An induced edge label is the absolute value of the difference between the two end vertex labels. Graceful labeling have applications in coding theory, x-ray, crystallography, radar, astronomy, circuit design and communication networks, addressing and data base management A complete and current summary of graceful and non-graceful results along with some unproven conjectures can be found in Gallian’s dynamic survey [1] of graceful labeling. Solairaju and Ambika [2] have proved that the n copies of cycles C is graceful. Solairaju and Ambika [3] have showed that the connected graph E * Sn+1 (copies of k number of stars) is a graceful graph. Also they [4] obtained a result that nC ( n number of C) and its mirror image are graceful. Similarly they got double nC is graceful. Solairaju and Ambika [5] have showed that a unicycle graph from Copies of Stars on cycles is a graceful graph.Solairaju and Ambika [6] have showed that the connected graph P and Mirror image of P are graceful. Also they [7] obtained a result -Tree and nare graceful graph. Preliminaries Definition 2.1 The graceful labeling of a graph G with q edges is a function f : V(G)such that distinct vertices receive distinct numbers and { f(v) - f(u) : uv E(G)}={1,2,3,…,q}. A graph is graceful if it has a graceful labeling. Definition 2.2 A path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path with n vertices is denoted by PDefinition 2.3 The graph 2[P] is defined as a connected simple graph, which contains the vertex set V= { v2n+5} with (2n+6) vertices and edges q = (2n+5) such that v is adjacent to {vn+5,n+6 , 2n+5 is adjacent to {v3, n+2}. Also n+3 are connected. The arbitrary labeling of the graph 2[P] is shown in the following figure 1. Definition 2.5 The graph [C] is defined as a connected simple graph, which contains the vertex set V= { v2n+2} with (2n+3) vertices and edges q = (2n+3) such that v is adjacent to v is adjacent to {vn+2is adjacent to {vn+3, n+4,n+52n+2}.The arbitrary labeling of the graph [C] is shown in the following figure 3. Figure 3: The graph [CDefinition 2.6 The graph [3P ] is defined as a connected simple graph, which contains (4n+12) vertices and edges q = (4n+11) such that yis adjacent to y and is adjacent to yand S contains the vertices {u/ i=1,2,3,…,n}and is adjacent to yand S contains the vertices {v/ i=1,2,3,…,n}and is adjacent to y/ i=1,2,3,…,n}and is adjacent to y11 / i=1,2,3,…,n}and is adjacent to yThe arbitrary labeling of the graph [3P* 4S ] is shown in the following figure 4. 0 2n+1 2n+2 V n+3 1 2 V3     n+1 n+2 Define f : { 0,1,2,…,q} by f(v) = i , where i = 0,1,2,…,q and q = (2n+5) By usual way to graceful of the graph 2[Pe(uv) = is edge labeling of the graph. The graph 2[P] is graceful from labeling f and e (2.3) Example: 3.2 The connected graph 2[P] is graceful. Gracefulness of the graph with 15 edges such that the vertex set V= {v2,…,labeled in the following manner which is shown in figure 5. Figure 5: The graph 2[PTheorem 3.3 The connected graph 2[P] is graceful. Proof: The graph 2[P] is a connected graph followed from definition 2.4 The graph 2[P] has (2n+8) vertices and (2n+7) edges. The graph 2[P] has an arbitrary labeling of vertices as in figure 2. Define f : { 0,1,2,…,q} by f(v) = i , where i = 0,1,2,…,q and q = (2n+7) By usual way to graceful of the graph 2[Pe(uv) = is edge labeling of the graph. The graph 2[P] is graceful from labeling f and e (2.4) Example: 3.4 The connected graph 2[P] is graceful. Gracefulness of the graph with 17edges such that the vertex set V= {v2,…,labeled in the following manner which is shown in figure 6. 4 2 7 5         To be graceful of above graph, define a map f: V{0,1,2,…,q} where q = 2n by ) = 1 ; f(v) = q ; ) = q –2i+1 for i =1,2,3,…,(n-1) Also f(v) is the absolute difference of f(v) and f(v). The graph double star K1,n,n is graceful. Hence this labeling is graceful. Example: 3.6 The double star K1,6,6 is graceful. Gracefulness of the graph with 12 edges such that the vertex set V={v} is labeled in the following manner which is shown in figure 8 Figure 8: The graph K1,6,6 Theorem 3.7 The triple star K1,n,n,n is graceful. Proof: Gracefulness of the graph K1,n,n,n with (3n) edges such that the vertex set } is labeled in the following manner which is shown in figure 9 3 0 2 4 6 8 Theorem 3.9 The connected graph [C] is graceful Proof: The graph [C] is a connected graph followed from definition 2.5 The graph 3* 2Sn] has (2n+3) vertices and (2n+3) edges. The graph [C] has an arbitrary labeling of vertices as in figure 3.Define f : V(G) { 0,1,2,…,q} where q = (2n+3) by ) = 0 ; f(v2n+3) = 2i-1 ; f(v) = 2i for i=1,2,3,…,n. By usual way to graceful of the graph [Ce(uv) = is edge labeling of the graph. The graph [C] is graceful from labeling f and e (2.5) Example: 3.10 The connected graph [C] is graceful Gracefulness of the graph with 13 edges such that the vertex set V={v} is labeled in the following manner which is shown in figure 11. Figure 11: The graph [CTheorem 3.11 The connected graph [3P] is graceful. Proof: The graph [3P] is a connected graph followed from definition 2.6. The graph [3P] has (4n+12) vertices and (4n+11) edges. The graph [3P] has an arbitrary labeling of vertices as in figure 4.Define f : V(G) { 0,1,2,…,q} where q = 5 7 9 References [1]J. A. Gallian. A Dynamic Survey of Graph Labeling. The Electronic Journal of [2]A. Solairaju and S. Ambika, A new class of graceful graphs, Antarctica. J. Math., 5(2)(2008), 65-76 [3]A. Solairaju and S. Ambika, Gracefulness of a new class from copies of stars, communicated to International Journal of computer application, USA. [4]A. Solairaju and S. Ambika, Gracefulness of a new class of mirror image of and mirror image of copies of double nC[5]A. Solairaju and S. Ambika, Gracefulness of a unicycle graph from Copies of Stars on Cycles, communicated to Electronic Notes in discrete Mathematics in [6]A. Solairaju and S. Ambika, Gracefulness of a new class of p and mirror image of p communicated. [7]A. Solairaju and S. Ambika, Gracefulness of a new class from Copies of ETree and n3, communicated.