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^{1}Department of Mathematics, Pondicherry University, Pondicherry, India

^{2}Department of Mathematics, Vallam University, Tamil Nadu, India

- *Corresponding Author:
- B. Kavitha Department of Mathematics, Pondicherry University Pondicherry, India;
**Email:**[email protected]

**Received:** 03-Mar-2022, Manuscript No. JSMS-22-55261; **Editor assigned:** 07-Mar-2022, Pre QC No. JSMS-22-52261(PQ); **Reviewed:** 21-Mar-2022, QC No. JSMS-22-55261; **Revised:** 02-May-2022, Manuscript No. JSMS-22-55261 (R); **Published:** 17-May-2022, DOI: 10.4172/JSMS.8.5.006.

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A graph G=(V(G), E(G)) with vertex set V is said to have a prime labeling if its vertices can be labeled with distinct positive integer 1,2,3 V such that for edge u v E(G), the labels assigned to u and v are relatively prime. A graph which admits prime labeling is called a prime graph. Graph labeling is an important area of research in Graph theory. There are many kinds of graph labeling such as graceful labeling, Magic labeling, Prime labeling, and other different labeling techniques. In this paper we discuss prime labeling for some graphs.

We also discuss prime labeling in the related of some graph operations namely cycle, path, crown, Fan, star and wheel graph.

Labeling; Prime labeling; Prime labeling of path graph; Cycle graph; Crown graph; Fan graph; Star graph; Wheel graph

In this article, we consider only finite simple undirected graph [1]. The graph G has vertex set V=V(G) and edge set E=E(G). The labeling of a graph G is an assigning of integers either to the vertices or edges or both subject to certain conditions. The notion of a prime labeling was introduced by Roger Entringer and was discussed in a paper [2] for notations and terminology, [3]. Many researchers have studied prime graph for example in Fu [4]. H has proved that the path Pn on n vertices is a prime graphhave proved that the Cn on n vertices is a prime graph [5]. We refer to have proved Edge Vertex Prime Labeling for Wheel, Fan and Friendship Graph [6]. have proved that wheel Wn is a prime graph [7]. have proved the prime labeling for some Fan related graphs [8]. For latest survey on graph labeling, we refer to [9] have proved the prime labeling for some cycle related graphs [10] have proved the Prime labeling for some fan related graphs. The following definitions and notations are used in main results [11].

- Let G = (V(G), E(G)) be a graph with p vertices. A bijection f: V(G) → {1, 2, … p} is called a prime labeling if for each edge e = {u, v} belongs to E, we have gcd{f(u), f(v)} =1. A graph which admits prime labeling is called a prime graph.

- A simple graph of ‘n’ vertices(n≥3) and n edges forming a cycle of length ‘n’ is called as a cycle graph. In a cycle graph, all the vertices are of degree is 2. By adding the path, the new vertices of v
_{1}, v_{2}, … v_{m}, and the new graph G is denoted by C_{n}@P_{m}

- The crown graph on 2n vertices is an undirected graph with two set of vertices {u1, u2, … un} and {v1, v2, … v n} and with on edge from ui to vj whenever i ≠ j. By adding the path, the new vertices of w1, w2, … w m, and the new graph G is denoted by crown C
_{n}@P_{m}

- The Friendship graph F n is a graph which consists of ⴄ −triangles with a common vertex. If V(G) = 2n+1 and E(G) = 3
_{n}by adding the path, the new vertices of v_{1}, v_{2}, … v_{m}, and the new graph G is denoted by F_{n}@P_{m}

- The star graph S
_{n}is special type of graph in which n-1 vertices have degree 1 and single vertex have n-1 degree. This look like n-1 vertex is connected to central vertex. A star graph wit total n vertex is termed as S_{n}. By adding the path, the new vertices of v_{1}, v_{2}, … v_{m}, and the new graph G is denoted by S_{n }@P_{m}

- The wheel graph W
_{n}is obtained by joining all vertices of a cycle C_{n}to a further vertex is called center. If V(G) = n+1 and E(G) = 2n by adding the path, the new vertices of v1, v2, … v m, and the new graph G is denoted by W_{n}@P_{m}

- Gear graph G
_{n}also known as a bipartite wheel graph is a wheel graph with a vertex added between each pair of adjacent vertices of the outer cycle. Gear graph G_{n}has 2r+1 vertices and 3r edges. By adding the path, the new vertices of w1, w2, … w m, and the new graph G is denoted by G_{n}@Pm

**Theorem**

The cycle and path graph are a prime graph. Then the graph Cn @ Pm prime labeling of the graph

**Proof: **

Let G be the graph obtained by joining cycle Cn and a path Pm, then the graph C_{n} @ P_{m} admit to prime labeling of the graph.

Let u1, u2, … un be the vertices of cycle C_{n }and v1, v2, … vn be the vertices of path P_{m}.

Clearly vertex labels are distinct. Then f admits prime labeling. Thus, C_{n} @ P_{m} is a prime graph.

Prime labeling of graph C_{4} @ P_{6}

The prime labeling of the graph is presented in the following graph.

**Theorem**

The crown and path graph are a prime graph. Then the graph C_{n} @ P_{m} prime labeling of the graph

**Proof: **

Let G be the graph obtained by joining crown C_{n} by a path Pm admit to prime labeling of the graph.

Let u1, u2, … un be the vertices of crown C_{n} and v1, v2, … v_{n} be the vertices of cycle C_{n}, then

w1, w2, … w_{n} be the vertices of path Pm.

Clearly vertex labels are distinct. Then f admits prime labeling. Thus, C_{n} @ P_{m} is a prime graph.

Prime labeling of graph crown C_{3} @ P_{4}

The prime labeling of the graph is presented in the following graph.

**Theorem**

The friendship and path graph are a prime graph. Then the graph F_{n} @ P_{m} prime labeling of the graph

**Proof: **

Let G be the graph obtained by joining friendship F_{n} by a path Pm admit to prime labeling of the graph.

Let u1, u2, … u_{n} be the vertices of friendship F_{n} and v1, v2, … v_{n} be the vertices of path P_{m}.

Clearly vertex labels are distinct. Then f admits prime labeling. Thus, F_{n} @ P_{m} is a prime graph.

Prime labeling of graph F_{3} @ P_{6}

The prime labeling of the graph is presented in the following graph.

**Theorem**

The star and path graph are a prime graph. Then the graph S_{n} @ P_{m} prime labeling of the graph

**Proof: **

Let G be the graph obtained by joining star Sn by a path Pm admit to prime labeling of the graph.

Let u0, u1, u2, … u_{n} be the vertices of star Sn and v1, v2, … v_{n} be the vertices of path P_{m}.

Clearly vertex labels are distinct. Then f admits prime labeling. Thus, S_{n} @ P_{m} is a prime graph.

Prime labeling of graph S_{8} @ P_{5}

The prime labeling of the graph is presented in the following graph.

**Theorem**

The wheel and path graph are a prime graph. Then the graph W_{n} @ P_{m} prime labeling of the graph

**Proof: **

Let G be the graph obtained by joining wheel Wn by a path P_{m}, admit to prime labeling of the graph.

Let u0, u1, u2… u_{n} be the vertices of wheel Wn and v1, v2…v_{n} be the vertices of path P_{m}.

Clearly vertex labels are distinct. Then f admits prime labeling. Thus, W_{n} @ P_{m} is a prime graph.

Prime labeling of graph W_{6} @ P_{5}

The prime labeling of the graph is presented in the following graph

**Theorem **

The Gear and path graph are a prime graph. Then the graph G_{n} @ P_{m} prime labeling of the graph

**Proof:**

Let G be the graph obtained by joining Gear G_{n} by a path P_{m}.

Let u1, u2, …and v1, v2, …v_{n} be the vertices of Gear G_{n} and w1, w2, … w_{n} be the vertices path P_{m}.

Clearly vertex labels are distinct.

Then f admits prime labeling. Thus, G_{n} @ P_{m} is a prime graph.

Prime labeling of graph G_{6} @ P_{5}

The prime labeling of the graph is presented in the following graph.

Prime labeling has been studied for then five decades. A huge number of research articles published in the area of graph theory and discrete mathematics. In this paper, we studied the prime labeling of some graph for, C_{n}@P_{m}, F_{n}@P_{m}, crown C_{n}@P_{m}, S_{n}@P_{m} and W_{n}@P_{m}, G_{n}@P_{m} in necessary conditions, In future work for some connected graphs.

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