Let G be a graph on n vertices. A bijective function f : V (G) → {1, 2,…,n} is said to be a prime labeling if for every e = xy, GCD{f (x),f (y)} = 1. A graph which permits a prime labeling is a “prime graph”. On the other hand, a graph G is a prime distance graph if there is an injective function g : V(G) → Z (the set of all integers) so that for any two vertices s & t which are adjacent, the integer |g(s) – g(t)| is a prime number and g is called a prime distance labeling of G. A graph G is a prime distance graph (PDL) iff there exists a “prime distance labeling” (PDL) of G. In this paper, we obtain the prime labeling and prime distance labeling of certain classes of graphs.
Graph theory plays a significant role in a variety of real-world systems. Graph concepts such as labeling and coloring are used to depict a variety of processes and relationships in material, social, biological, physical, and information systems. Specifically, graph labeling is used in communication network addressing, fault-tolerant system design, automatic channel allocation, etc. 2-odd labeling assigns distinct integers to the nodes of G(V, E) in such a manner, that the positive difference of adjacent nodes is either 2 or an odd integer, 2k ± 1, k ∈ N . So, G is a 2-odd graph if and only if it permits 2-odd labeling. Studying certain important modifications through various graph operations on a given graph is interesting and challenging. These operations mainly modify the underlying graph's structure, so understanding the complex operations that can be done over a graph or a set of graphs is inevitable. The motivation behind the development of this article is to apply the concept of 2-odd labeling on graphs generated by using various graph operations. Further, certain results on 2-odd labeling are also derived using some well-known number theoretic concepts such as the Twin prime conjecture and Goldbach's conjecture, besides recalling a few interesting applications of graph labeling and graph coloring.
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