Group $S_{3}$ cordial prime labeling of graphs

The process of assigning the binary numbers 0 and 1 to the edges of a graph \(G = (V, E)\) through evaluating functions defined on the vertex set V and the edge set E of G using the concepts of prime and Narayana numbers by satisfying cordiality on the edges. This process on the graph G is known as Narayana prime cordial labeling of G and this graph G is called Narayana prime cordial graph. In this article, we compute the Narayana prime cordial labeling of Web graphs and Flower graphs.

Let G be a graph. A bijection f:V→ {1,2,…..|V|} is called a prime labeling [3] if for each edge e=uv in E, we have GCD{ f(u),f(v)}=1. A graph that admits a prime labeling is said to be a prime graph. In this paper we show that bull graph admits Prime labeling in the context of variety graph operations namely duplication of vertex, fusion of vertices and Switching in Bull graph.

A graph with a vertex set is said to be a prime graph if there exists a bijective mapping , where denotes the number of vertices in , such that for any two adjacent vertices and in have . Tensor Product graph is a way to combine (compose) two graphs into one larger and more complex graph. The result is a new graph that reflects the connection properties of the two original graphs, but in a very specific and more complex way than other graph operations. Therefore, this research aims to determine whether there is prime labeling in the class of graphs resulting from the Tensor Product of the path graph and the cycle graph . The research employed analytical and exploratory methods with a trial-and-error strategy to determine the labeling that possesses a prime property. The results of this study prove that two classes of the Tensor Product graph for , and graph , for are prime graph. This finding expands the results on classes of graphs that admit prime labeling and provides a basis for further research on graph labeling in other graph operations

A graph H is a prime distance graph if there exists an one-to-one function h : V (H ) \(\to\) Z such that for any two adjacent vertices x and y, the integer |h(x) - h(y)| is a prime. So H is a prime distance graph if and only if there exists a prime distance labeling of H. If the edge labels of H are also distinct, then h becomes a distinct prime distance labeling of H and H is called a distinct prime distance graph. The generalized Petersen graphs P (n, k) are defined to be a graph on 2n (n \(\ge\) 3) vertices with V (P (n, k)) = {vi , ui : 0 \(\le\) i \(\le\)n - 1} and E(P (n, k)) = {vi vi+1 , vi ui , ui ui+k : 0 \(\le\) i \(\le\) n - 1, subscripts modulo n}. In this chapter, we give a concise overview on these two labeling, specifically, highlight that the generalized Petersen graphs P (n, 3) permit a prime distance labeling for all even n > 5 and conjecture that P (n, 2) and P (n, 3) admit a prime distance labeling for any n \(\ge\) 5 and all odd n \(\ge\) 5, respectively. Moreover, the cycle Cn admits a distinct prime distance labeling for all n \(\ge\) 3.

Research in graph theory has lead to one of the important area called labeling of graphs. There are different types of labeling such as graceful labeling, magic labeling, edge-graceful labeling, prime labeling, radio labeling, harmonious labeling etc., In this paper, we investigate the existence of edge graceful labeling for the Corona graph Cn⊙K2¯ (n is odd n ≥ 3). Also we have defined a flower graph which does not admit an edge-graceful labeling.

kth Fibonacci Prime Labeling is defined as labeling the vertices of a graph with distinct Fibonacci numbers starting since the kth Fibonacci term sustaining the condition that the , where and are labels of any adjacent vertices u and v. Graphs formed by consecutively connecting identical base graphs, linearly or in alternating pattern, is called Snake graph. In this paper, we show that some snake graphs admit kth Fibonacci prime labeling.

We discuss vertex prime labeling of graphs obtained by using Shel graph. We have obtained vpl of Shel graphs, antena of Shel, Path union of Sn (n =4), snake of Shel and one point union of Shel, star of shel and a few other.

In this paper, we discuss about the prime labeling of Split graph of Coconut tree CT(m,n).Here m is the number of vertices in the path and n is the number of pendent vertices in the Coconut tree.A graph G with vertex set V is said to have a prime labeling if its vertices are labeled with integer 1,2,….,|V|such that for every edge xy the labels assigned to x and y are relatively prime.A graph which admits prime labeling is called a prime graph.