Some Results on Prime Cordial Labeling of Lilly Graphs

A PCL of G is a bijective map g from V to {1, 2, 3, |V|} in such a way that if an edge st is given label 1 if GCD(g(s), g(t)) = 1 & 0 otherwise, then the edges given 0 & 1 differ by at most 1 i.e; |eg (0) − eg (1)| ≤ 1. If a graph permits a PCL, then it is called a PCG. In this paper, we prove that lilly graph admits a PCL. Further, we have shown that lilly graph under some graph operations like switching of a vertex, duplication of a vertex, degree splitting graph and barycentric subdivision admits a PCL which may find its application in the development of artificial intelligence.


Introduction
Graph labeling is a widely used and fastest growing area in the field of mathematics and computer science. Now a days, when data security is a major area of concern, various researchers and scientist are working to develop the techniques and softwares that can resolve the issues. Graph labeling is used in many areas of science and technology. A lot of graph labeling techniques are discussed in [4], we enlist a few of them which are finding their use in different aspects of artificial intelligence [6].
1. Radio labeling is finding its application in fast communication in sensor networks. 2.The designing of fault tolerance system with particularized degree, facility graph is used. 3. The concept of chromatic number is widely used in solving many complex problems in computer which is also a type of graph labeling. 4. Mobile Adhoc Networks (MANETS) problems can also be resolved by using graph labeling. 5. Automatic routing with graph labeling is done when each network usually path, cycle, circuit, walk and connected graph repesent a fixed network and labeling is done with a constant which helps routing to involuntary detect the next node in the network. 6. Behavior trees are used in robotics.
For number theory concepts, refer to [2] and for terms and terminology related to graph theory that have not been defined here, we refer to Hararay [5]. For detail survey on various  [4]. Cahit [3] is the introducer of cordial labeling. For the sake of simplicity, by 'PCL' we mean a prime cordial labeling and by 'PCG' we mean a prime cordial graph.

Basic Definitions and Results
In this section, we discuss the PCL of lilly graphs. First we recall some basic definitions for the sake of completeness.
After Cahit [3], various researchers introduced a lot of graph labeling techniques with some type of restrictions and/or variations in the cordial theme. The notion of PCL was introduced by Sundaram, Ponraj & Somasunaram [8].   Trees constitute an important class of graphs in graph theory. Many researchers are working on trees for different kind of graph labeling. J. Baskar Babujee in [1] proved that full binary tree admits a prime cordial labeling. Motivated by [1], [7], and [9], we attempt to contribute some results on particular type of a tree namely lilly graph.
Keeping above in view, these edges will contribute (bear) 0 and the rest of the edges of I n will contribute 1 (since the gcd of their end vertices is equal to 1). Evidently, e f (0) = e f (1) = 2n − 1 which shows that I n is a PCG.  will not able to be labeled, since there are exactly 4n 2 − 1 number of even labels available. For unlabeled star pendant vertices we assign the labels 3 and 9.
Observe gcd(f (u 3n ), f (u i )) > 1, ∀ 1 ≤ i ≤ 2n, gcd(f (u 3n ), f (u 3n+1 )) > 1 and gcd of f (u k ) with all pendant vertices of star except for the those that are lebeled with 3 and 9, will be greater than 1. Clearly there are exactly 4n − 3 number of edges that will bear 0 labels and for the rest of the edges, the gcd of their end vertices will be equal to 1.
Case 2: When we switch pendant vertices of path in lilly graph. i.e; u 2n+1 or u 4n−1 . Without loss of generality, let us switch u 2n+1 Assign the unused labels out of {1, 2, ..., 4n − 1} in the increasing order. Observe The edges formed using these vertices will bear 0 label which are 4n − 3 in count. The gcd of the end vertices of the rest of the edges is qual to 1. Evidently, e f (0) = e f (1) = 4n − 3 which justifies |e f (0) − e f (1)| ≤ 1. We see in both cases that lilly graph is invariant under the graph operation of switching of any pendant vertex for PCL.
Assign unused labels to the remaining vertices in any order. Observe that gcd(f (u i ), f (u i+1 )) > 1 for 2n + 1 ≤ i ≤ 3n − 2 and for 3n + 1 ≤ i ≤ 4n − 2, and for rest of the edges, since the gcd of their end vertices is equal to 1, therefore those edges will be labeled with label 1. Clearly, e f (0) = e f (1) = 2n − 4. Therefore G is a PCG.
Theorem 4. Duplication of apex vertex in I n admits a PCL for n ≥ 3.
Proof. Let G 1 be obtained by duplicating the apex vertex of I n namely u 3n , by a vertex namely v.
The gcd of the end vertices of the remaining edges of G is equal to 1. we find that e f (0) = e f (1) = 3n which proves that G is prime cordial.  Figure 5. Switching of u 9 in lilly graph I 3 Figure 6. Duplication of u 3n in I 4 Theorem 5. The duplication of any pendant vertex in I n for n ≥ 2 permits a PCL.
The edges due to above vertices will bear 0 label. For the rest of the edges-since the gcd of their end vertices is equal to 1 therefore, those edges shall be labeled with label 1. Evidently, e f (1) = 2n and e f (0) = 2n, which proves that G is prime cordial.   Figure 8. Duplication of u 12 in I 5 Assign unused labels out of the available labels to the unlabeled vertices in increasing order with respect to the increasing order of indices.

Conclusion
We have shown the PCL of lilly graph with various graph operations namely switching a vertex, duplication of vertex by a vertex, degree splitting graph and barycentric subdivision. Observe 10 that lilly graph is a type of tree, so it is an interesting task to investigate PCL for more tree graphs. Further there is a scope for studying the prime cordial labeling of lilly graph with some other graph operations. We believe that the concept of PCL may play a vital role in the area of robotics and artificial intelligence which is for the future work.