PRIME LABELING OF SPECIAL GRAPHS

Prime labeling is the most interesting category of graph labeling with various applications. A graph with vertices are said to have prime labeling if its vertices are labeled with distinct positive integers such that for each edge the labels assigned to and are relatively prime, where and are vertex set and edge set of , respectively. Therefore, the graph has a prime labeling whenever any of two adjacent vertices can be labeled as two relative prime numbers and is called a prime graph. In our work, we focus on the prime labeling method for newly constructed graphs obtained by replacing each edge of a star graph by a complete tripartite graph for and , which are prime graphs. In addition to that, investigate another type of simple undirected finite graphs generalized by using circular ladder graphs. These new graphs obtained by attaching at each external vertex of the circular ladder graph and proved that the constructed graphs are prime graphs when and . Finally, focus on another particular type of simple undirected finite graph called a scorpion graph, denoted by . The Scorpion graph gets its name from shape, which resembles a scorpion, having vertices are placed in the head, body, and tail respectively. To prove that the scorpion graph has prime labeling, we used two results that have already been proved for ladder graphs.


INTRODUCTION
Graph labeling is a prominent research area in Graph theory, and there are considerable number of open problems and literature are available for various types of graphs. Rosa introduced the theory behind graph labeling in the 1960s. Roger Entringer introduced the concept of prime labeling, and around 1980 he conjectured that all trees have prime labeling, which is not settled until today. Tout. et al. introduced the method of prime graph labeling in 1982. In prime graph labeling, distinct positive integers are assigned to the vertices which are less than or equal to the number of vertices in the graph such that labels of adjacent vertices are relatively prime. Edge labeling is another particular area in graph labeling and was introduced by Deretsky, Lee, and Mitchem in 1991. Throughout this paper, we considered the vertex prime labeling for a new type of graphs. Many researchers had proved that various types of graphs are prime graphs. The graph with vertex set and edge set with vertices is said to have prime labeling if there exists a bijective mapping such that for each edge in , and are relatively prime and such a graph is called a Prime graph. In our previous work, prime labeling of Crab graph, Roach graph, Centerless double wheel graph, and a Complete tripartite graph for have been discussed. In this paper, we consider prime labeling of newly constructed graphs obtained by replacing every edge of a star graph by the tripartite graph for and , Stripe Blade Fan Graph , simple undirected finite graph obtained by taking the union of the star graph and the circular ladder graph and the Scorpion graph. We provide a brief summary of definitions and theorems which are necessary for the present investigations.

MATERIAL AND METHODS
In our research, prime labeling of some special types of graphs were given by the following theorems. Then, the new vertex set is , for .
Also, the new edge set is , for .
So .
Define a function as follows: for .
for .
for .
Note that, for .
for .
for .
Consider, for , Therefore, vertex labels are distinct. Thus, labeling defined above gives a prime labeling for . Therefore, is a prime graph.

Example 1
Theorem 2: The graph obtained by replacing every edge of a star graph by the tripartite graph is a prime graph, where .
Proof: This proof is similar to the proof of theorem . However, the function is defined as follows for these types of graphs. Let be a graph obtained by replacing every edge of a star graph by , where . Let the vertices of be with as the vertex at the center and every edge of is replaced by for where .  for , where .

Example 2
The prime labeling of the graph obtained by replacing every edge of a star graph by using labeling appears in the following figure. Clearly, vertex labels are distinct. Thus, the labeling method defined above gives a prime labeling for . Therefore, is a prime graph. This proof is similar to the proof of Theorem .

Example 3 When
; the resultant graph is called a Stripe Blade Fan Graph . The following figure shows stripe blade fan graphs for and . Then, and .  Clearly, vertex labels are distinct. Thus, the above labeling method gives a prime labeling for . Hence, is a prime graph.  Proof: By using Theorem in [ ], vertices have prime labeling, which are in the head and body of the scorpion, when is prime. Let the starting vertex in the tail is placed at between and vertices, and we claim that the following vertex labeling ( Figure  3) gives a prime labeling: Let , then and , that implies . Since divides both and , the only possible value for is ( and are consecutive positive integers). Also, all vertices in the tail are relatively prime because of consecutive positive integers. Thus, the scorpion graph is a prime graph whenever is prime.

Theorem 7:
If is prime, then the Scorpion graph has a consecutive cyclic prime labeling with the value assigned to vertex , where and .
Proof: This proof is the same as the above proof. Here, we only consider the starting point of the tail in the scorpion graph. By using Theorem in [ ], vertices have prime labeling, which is in the head and body of the scorpion when is prime. Let the starting vertex in the tail is placed in between and vertices, and we claim that the following vertex labeling gives a prime labeling: Since vertices have prime labeling, which is in the head and body of the scorpion, it suffices to check only the vertex labels arising from the endpoints of the following particular edges: the vertical edge connects vertex labels and , and the vertical edge is attaching vertex labels and .
Let . Then and , implies Since divides both and , the only possible value for is 1 ( and are consecutive positive integers).

Let
, then and that implies . Since divides both and , the only possible value for is 1 ( and are consecutive positive integers). Also, all vertices in the tail are relatively prime because of consecutive positive integers. Thus, the Scorpion graph is a prime graph whenever is prime.

RESULTS AND DISCUSSION
The field of graph theory plays a vital role in discrete mathematics. Graph labeling is one of the important areas of graph theory. Here we present prime labeling of newly constructed graphs in Theorem to Theorem . In Theorem and Theorem , we discuss the prime labeling method of graph obtained by replacing every edge of a star graph by the tripartite graph . Using the first three examples, it has been shown that the resulting graph has prime labeling when and (Fig.2, 3, and  4). In addition to that, the stripe blade fan graph can be occurred when in .Further, the following conjecture can be obtained for the stripe blade fan graph.
Stripe Blade Fan graph conjecture: Stripe blade fan graph has prime labeling for all finite . It has been noticed that once we label vertices with an integer from to with assigning to the center vertex. The integers between and for , there is always a prime number in each one-third interval. Then that number can be assigned to the end of the fan blade. Since other numbers are relatively prime with these prime numbers, the stripe blade fan graph is a prime labeling graph.
Furthermore, another newly constructed graph was introduced by taking the union of circular ladder graph and star graph of . Prime labeling of this graph is given by Theorem . It has been illustrated for the case when by figure . Finally, we had proved that the scorpion graphs have prime labeling when and are prime, where and . It is very interesting to investigate prime labeling of the graphs that have structures of some insects and small animals in the real world.

CONCLUSION
Prime labeling is the most valuable area of graph labeling with various applications. The prime numbers and their behavior are of great importance as prime numbers are scattered, and there are arbitrarily large gaps in the sequence of prime numbers. It has been shown that newly constructed graphs obtained by replacing every edge of a star graph with the tripartite graph for and . According to the given results, the graphs obtained by replacing every edge of a star graph by for and are prime graphs where . Also, obtained a prime labeling method for the graphs obtained by attaching at each external vertex of the circular ladder graph is more difficult when and As a future work, we are planning to generalize these results for scorpion graphs that have walking legs. We would also further investigate additional structures which have prime labeling.