Proceeding2562

1489 การประชุมวิชาการระดับชาติมหาวิทยาลัยทักษิณ ครั้งที่ 29 ประจ�ำปี 2562 วิจัยและนวัตกรรมเพื่อการพัฒนาที่ยั่งยืน GRACEFUL LABELING FOR DUPLICATION OF A VERTEX ON THE n P BY A NEW EDGE Siwaporn Saewan 1* and Thanapa Khawman 2 Abstract Introduction : Let ( ( ), ( )) G V G E G = be a graph with vertex set ( ) V G and edge set ( ) E G . For u and v be the vertices if , e uv = then u and v are end vertices of the edge e . Two vertices u and v are adjacent if they are join by an edge. The degree of a vertex v is a number of edges which v is an end vertex. A pendent vertex is a vertex have degree 1. A path 1 n v v − in a graph G is a finite sequence of vertices 1 2 3 1 , , ,..., , ,..., i i n v v v v v v + such that 1 i i v v + is an edge of G and any vertex visited at most once. u and v are connected if there exists a u v − path. A graph G is connected if all the vertices are connected to each other. Let G be a graph with q edge, a graceful labeling of a graph G is a one to one function : ( ) {0,1, 2,..., } f V G q → such that { ( ) ( ) : ( )} {1, 2,3,... }. f u f v uv E G q − ∈ = A graph which has a graceful labeling is called a graceful graph. A path graph n P is a connected graph on n vertices such that each vertex has degree at most 2. Objective : 1) To establish two new graphs by duplication of a pendent vertex in a Path n P by a new edge and by duplication of a middle vertex in a Path n P by a new edge. 2) To find a graceful labelling of new graphs. Methods : Studying and investigating on graph operation. Studying and investigating on graceful labeling. We find a graceful labeling of new graphs. Results : The graph obtained by duplication of a pendent vertex in a Path n P by a new edge is graceful. The graph obtained by duplication of a middle vertex in a Path n P by a new edge is graceful. Conclusion : In this paper we show that the graph obtained by duplication of a pendent vertex in a Path n P by a new edge and the graph obtained by duplication of a middle vertex in a Path n P by a new edge are graceful. Keywords : Graceful Labeling, Duplication of a Vertex, Path