One Raised Product Prime Labeling of Some Star Related Graphs

One raised product prime labeling of a simple. finite and un direct graph G with p vertices and q edges is the labeling of the vertices of the graph with first p natural numbers, edges with product of the labels of the end vertices plus one. If the greatest common divisor of each vertex of degree greater than one is one then the graph is called one raised product prime graph. Here the greatest common divisor of the labels of the edges incident on a vertex is called greatest common incidence number of that vertex. Here we proved that star, bistar, splitting graph of a star, shadow graph of a star , jelly fish graph and some more star related graphs admit one raised product prime labeling. Keywords— Graph labeling, product, prime labeling, prime graphs, star


INTRODUCTION
The concept of one raised product prime labeling was introduced in [5] by Sunoj B S and Mathew Varkey T K. Later in [6] and [7] they extend the study and proved the result for some path related graphs and some snake graphs. In this paper we proved that star, bistar, graph obtained by duplicating the apex vertex of star by a vertex, graph obtained by duplicating an edge of a star by an edge, graph obtained by duplicating the apex vertex of star by an edge, splitting graph of a star, shadow graph of a star, jelly fish graph and tensor product of star with path of length one admit one raised product prime labeling.
The paper contains three sections in which section I contains introduction to one raised product prime labeling, section II contains preliminaries and notations and section III contains main results and illustrations and section IV contains concluding remarks.

II. PRELIMINARIES AND NOTATIONS
In this paper we consider only those graphs which are simple finite and undirected. Graph is denoted by G, vertex set is denoted by V, edge set is denoted by E , p is the number of vertices of the graph G and q is the number of edges of the graph G.  V(G) = number of vertices of graph G,  E(G) = number of edges of graph G.

Definition 2.1[3]
If the vertices and edges of the graph are assigned values subject to certain conditions is known as graph labeling.

Definition 2.2[8]
The greatest common incidence number of a vertex is define as the greatest common divisor [1] of the labels of the edges incident on that vertex.

Definition 2.3[5]
Let G = (V(G),E(G)) be a graph with p vertices and q edges . Define a bijection f : V(G) {1,2,---, p} by f(v i ) = i , for every i from 1 to p and define a 1-1 mapping : E(G) set of natural numbers N by = .The induced function is said to be one raised product prime labeling, if for each vertex of degree at least 2, the gcin of the labels of the incident edges is 1.

Definition 2.4[5]
A graph which admits one raised product prime labeling is called one raised product prime graph.

Definition 2.5 [2]
A tree with one internal node and n leaves is called star and is denoted by K 1,n .

Definition 2.6[4]
For a graph G the splitting graph is obtained by adding to each vertex v a new vertex vʹ such that vʹ is adjacent to every vertex that is adjacent to v in G. The resultant graph is denoted as Sʹ(G).

Definition 2.7 [3]
The shadow graph D 2 (G) of a connected graph G is constructed by taking two copies of G say Gʹ and Gʹʹ join each vertex vʹ in Gʹ to the neighbors of the corresponding vertices in Gʹʹ.

Definition 2.8 [3]
Let G be a cycle of length 4 with one chord. The graph obtained by joining the apex vertex of star K 1,n and star K 1,m to the vertices of degree2 is called jelly fish graph and is denoted by JF(m,n). Definition 2.9 [3] Tensor product of path P 2 with star K 1,n is two copies of star K 1,n+1 .
So, gcin of each vertex of degree greater than one is 1. Hence K 1,n , admits one raised product prime labeling.

IV. CONCLUSION
Every graph is not one raised product prime graphs. So it is very interesting to find graphs or graph families which admit one raised product prime labeling. Here 9 graphs related to star graph family are investigated. This can be extended to some other graphs.