The necessity for good and tenable communication systems has motivated researchers to develop mobile communication networks (MCN). On the other hand, the huge functionalities of the global system of mobile (GSM) communication have given an increasing number of users. As the subscribers grow, the necessity for efficient and productive planning of the limited frequency spectrum of the GSM is inevitable, especially in highly dense areas. Researchers have proposed various algorithms for frequency or channel allocation (CA), as the discussions about CA methods to resolve the various practical issues in CA are going on. The literature reveals that the “Manhattan distance” concept can be used in scheduling and optimization problems. Similarly, the same idea makes it possible to discover a more tenable telecommunication system with “ease of connectivity” among subscribers, even when many users are on a common channel. Graph labeling is the most interesting idea in graph theory that has numerous uses in different fields, particularly in communication networks. 2-odd labeling assigns distinct integers to the nodes of a graph $$G\left( {V,E} \right)$$ such that the positive difference of adjacent nodes is either 2 or an odd integer, $$2k \pm 1,\;k \in N$$ . The motivation behind the development of this article is to study the applications of graph theory in communication networks through the concept of 2-odd labeling in graphs. 

Studying the Applications of Graph Labeling in Satellite Communication Through 2-Odd Labeling of Graphs

Certain Results on Prime and Prime Distance Labeling 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 <img src=image/13427595_01.png> in such a manner, that the positive difference of adjacent nodes is either 2 or an odd integer, <img src=image/13427595_02.png>, <img src=image/13427595_03.png>. So, <img src=image/13427595_04.png> 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. 

Ajaz Ahmad Pir

Papers

Certain Results on Prime and Prime Distance Labeling of Graphs

2-Odd Labeling of Graphs Using Certain Number Theoretic Concepts and Graph Operations